School geometry objective evidence

Make geometric definitions testable — Geometric definitions, motion witnesses and theorem proofs

• Define the segment AB and ray AB for distinct points A and B. Use A=(1,0), B=(3,0) to give a point on the ray but not the segment and a point on the line but not the ray.Written reasoning: review required
• A learner defines parallel lines as “two drawn segments that do not meet.” Repair the definition and provide a coordinate counterexample to the learner’s test.Written reasoning: review required
• Define a circle and its filled disk of positive radius r about O. Explain the ambiguity in “the arc from A to B” for two distinct points on the circle, including antipodal endpoints.Written reasoning: review required
• Define perpendicular lines and an angle using rays. Explain why an apparently right corner in a stretched image is not a proof of perpendicularity, and give a distance-based check for a triangle with sides 5,12,13.Written reasoning: review required

Separate a drawing from a geometric argument — Euclidean and coordinate geometry foundations

• Which is a definition rather than a conclusion inferred from appearance?Automatically checked response; reasoning still matters
• Which assumption is needed to infer AC=AB+BC?Automatically checked response; reasoning still matters
• On segment AB of positive length, suppose M and N are both midpoints. Prove M=N using distance from A along the segment.Written reasoning: review required

Treat a motion as a function on the plane — Geometric definitions, motion witnesses and theorem proofs

• Describe reflection in x=2 geometrically and give its coordinate rule. Identify its fixed points and prove distance preservation using coordinate differences.Written reasoning: review required
• Compare translating by (2,0) then rotating 90° counterclockwise about the origin with performing those actions in the opposite order. Derive both coordinate rules and test (1,3).Written reasoning: review required
• Apply H(x,y)=(2x,y) to O=(0,0), U=(1,1), V=(1,−1). Show that it fails both distance and angle preservation without using a cosine law.Written reasoning: review required

Move a figure without changing its shape — Euclidean and coordinate geometry foundations

• Prove that translating every point by (u,v) preserves the distance between arbitrary points (x1,y1) and (x2,y2).Written reasoning: review required
• Show by an example that translating by (1,0) and rotating 90° counterclockwise about the origin give different results in different orders.Written reasoning: review required

Classify Euclidean isometries algebraically — Axiomatic Euclidean and non-Euclidean geometry

• Give a determinant-one affine map that is not an isometry, and verify the failure.Written reasoning: review required
• Prove Qx+b preserves distance when QᵀQ=I.Written reasoning: review required

Find the complete symmetry set — Geometric definitions, motion witnesses and theorem proofs

• How many reflection axes does a non-square rectangle have?Automatically checked response; reasoning still matters
• For A=(0,0),B=(5,0),C=(7,3),D=(2,3), construct a nonidentity rotational symmetry and explain why this parallelogram has no reflection symmetry.Written reasoning: review required
• Compare trapezoid T with vertices (−3,0),(3,0),(2,2),(−2,2) and U with vertices (0,0),(6,0),(4,2),(1,2). Identify all reflection symmetries and nonidentity rotations. Here trapezoid means exactly one parallel side pair.Written reasoning: review required
• For a regular hexagon, describe all rotations and reflection axes and justify that there are no additional ones.Written reasoning: review required
• A learner claims every diagonal of every rectangle is a reflection axis. Refute the claim using vertices (0,0),(4,0),(4,2),(0,2), and state when the diagonal claim does hold.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• A regular octagon has center O. Determine all its rotational symmetry angles modulo 360 degrees and count its reflection axes, explaining why there are no additional rigid symmetries. Contrast a rectangle of side lengths 3 and 7, including its diagonal-reflection claim.Written reasoning: review required

Use relations without assuming rotations commute with reflections — Abstract algebra I: groups and symmetry

• Under the convention Dₙ has order 2n, how many symmetries does a regular pentagon have?Automatically checked response; reasoning still matters
• Label regular n-gon vertices modulo n, let r(i)=i+1 and s(i)=−i. Prove srs=r⁻¹.Written reasoning: review required
• For n≥3 prove the 2n maps rᵃ and rᵃs are distinct.Written reasoning: review required

Treat a motion as a function on the plane — Geometric definitions, motion witnesses and theorem proofs

• Rotate (4,0) counterclockwise by 90° about (1,−2). What is the image y-coordinate?Automatically checked response; reasoning still matters
• Describe reflection in x=2 geometrically and give its coordinate rule. Identify its fixed points and prove distance preservation using coordinate differences.Written reasoning: review required
• Give inverse operations for a translation, a rotation and a reflection. Explain why a rigid motion maps a segment to a congruent segment rather than scattering its points.Written reasoning: review required

Move a figure without changing its shape — Euclidean and coordinate geometry foundations

• What is the reflection of (3,−2) across the y-axis?Automatically checked response; reasoning still matters
• Rotate (4,−7) by 90° counterclockwise about the origin. What is the image y coordinate?Automatically checked response; reasoning still matters

Reflections, rotations and glide reflections — Axiomatic Euclidean and non-Euclidean geometry

• Prove reflection in n·x=c, with unit n, has formula R(x)=x−2(n·x−c)n and is an isometry.Written reasoning: review required
• Prove two reflections in parallel lines n·x=a then n·x=b give translation 2(b−a)n.Written reasoning: review required

Construct and verify a motion witness — Geometric definitions, motion witnesses and theorem proofs

• Map A=(0,0),B=(2,0),C=(0,3) to A′=(−1,5),B′=(−1,7),C′=(−4,5) with an explicit rigid sequence. Verify every vertex.Written reasoning: review required
• Map A=(0,0),B=(3,0),C=(0,2) to A′=(1,1),B′=(4,1),C′=(1,−1). Give a witness and explain why translations and rotations alone cannot produce this labeled correspondence.Written reasoning: review required
• A proposed witness for the previous target translates by (1,1) first and then reflects in the x-axis. Compute all images and repair it without changing the operation order.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• Map A=(1,0),B=(4,0),C=(1,2) to P=(2,5),Q=(2,2),R=(0,5) using a reflection followed by a quarter-turn and a translation. Verify all three points, write the total coordinate function and prove it preserves every distance.Written reasoning: review required

Move a figure without changing its shape — Euclidean and coordinate geometry foundations

• Show by an example that translating by (1,0) and rotating 90° counterclockwise about the origin give different results in different orders.Written reasoning: review required
• Translate (−3,5) by vector (7,−2). What is the image x coordinate?Automatically checked response; reasoning still matters
• Rotate (4,−7) by 90° counterclockwise about the origin. What is the image y coordinate?Automatically checked response; reasoning still matters

Construct and verify a motion witness — Geometric definitions, motion witnesses and theorem proofs

• Map A=(0,0),B=(3,0),C=(0,2) to A′=(1,1),B′=(4,1),C′=(1,−1). Give a witness and explain why translations and rotations alone cannot produce this labeled correspondence.Written reasoning: review required
• A rectangle with vertices (0,0),(2,0),(2,1),(0,1) is claimed congruent, with corresponding corners, to (0,0),(4,0),(4,1),(0,1). Give an obstruction. Does preserving right angles suffice for congruence?Written reasoning: review required

Define triangle congruence through a witness — Geometric definitions, motion witnesses and theorem proofs

• A rigid motion sends the three vertices of a triangle to three target vertices in order. Explain why it sends the complete triangular region onto the target region, and why arbitrary three-point functions would not suffice.Written reasoning: review required
• For A=(0,0),B=(4,0),C=(0,3) and P=(1,1),Q=(1,5),R=(−2,1), construct the correct correspondence and witness. Explain why A→P,B→R,C→Q is impossible.Written reasoning: review required

Move a figure without changing its shape — Euclidean and coordinate geometry foundations

• Prove that translating every point by (u,v) preserves the distance between arbitrary points (x1,y1) and (x2,y2).Written reasoning: review required

Match triangles with sufficient evidence — Euclidean and coordinate geometry foundations

• Triangle ABC is congruent to triangle PQR in that order. BC=13/2 cm. Find QR in cm.Automatically checked response; reasoning still matters
• What does equal angle-angle-angle data establish for nondegenerate triangles?Automatically checked response; reasoning still matters

Define triangle congruence through a witness — Geometric definitions, motion witnesses and theorem proofs

• Starting from a rigid witness taking ABC to PQR, prove the corresponding sides and angles agree. Why is listing three equal angles alone not a converse proof of congruence?Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• For nondegenerate labeled triangles, prove that a rigid-motion congruence exists if and only if all corresponding sides and angles are equal. In the converse, explain why the side data alone suffice. Illustrate with aligned triangles having A=(0,0), B=(8,0), AC=√20 and BC=√52: find both possible third vertices.Written reasoning: review required

Match triangles with sufficient evidence — Euclidean and coordinate geometry foundations

• Triangle ABC is congruent to triangle PQR in that order. BC=13/2 cm. Find QR in cm.Automatically checked response; reasoning still matters
• If AB=AC, prove angles ABC and BCA are equal by comparing triangles ABC and ACB.Written reasoning: review required

Congruence axioms and SAS — Axiomatic Euclidean and non-Euclidean geometry

• In R² put A=(0,0), B=(c,0) with c>0. Show that AC=b and included angle θ in (0,π), with C above AB, determine C uniquely.Written reasoning: review required

Derive the three triangle criteria — Geometric definitions, motion witnesses and theorem proofs

• Prove SSS for two nondegenerate triangles by aligning a matching side of length c on the x-axis. Use lengths b from the origin and a from (c,0), and justify the possible reflection.Written reasoning: review required
• Prove SAS for two nondegenerate triangles with two corresponding side lengths and the included angle equal. Explain precisely why “included” is essential to the proof.Written reasoning: review required
• Prove ASA when the matched side lies between the matched angles. Explain why a degenerate configuration is outside this statement.Written reasoning: review required
• A learner claims SSA is proved by the SAS argument because two lengths and an angle are known. Refute the reasoning with A=(0,0), a ray along the positive x-axis for B, fixed C=(4,3), and BC=√10. Find both possible B positions and compare AB.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• Explain why ASA determines a unique triangle up to rigid motion when the data describe an existing nondegenerate triangle. Then test the competing SSA data A=(0,0), C=(6,4), B on the positive x-ray and BC=√20. Give both B positions and identify the equal data.Written reasoning: review required

Match triangles with sufficient evidence — Euclidean and coordinate geometry foundations

• Which data establish SAS congruence?Automatically checked response; reasoning still matters
• Explain geometrically why sides AB=c, BC=a and an angle at A need not determine a unique triangle when BC is opposite the given angle.Written reasoning: review required

Congruence axioms and SAS — Axiomatic Euclidean and non-Euclidean geometry

• In R² put A=(0,0), B=(c,0) with c>0. Show that AC=b and included angle θ in (0,π), with C above AB, determine C uniquely.Written reasoning: review required
• Explain why quoting the Euclidean cosine rule to prove SAS in neutral geometry risks circularity.Written reasoning: review required

Prove angle relations with parallel lines — Geometric definitions, motion witnesses and theorem proofs

• Prove vertical angles equal without assuming the lines are perpendicular or the drawing is symmetric.Written reasoning: review required
• Explain a derivation of equal alternate interior angles for parallel lines using a half-turn about the midpoint of the segment joining their intersections with a transversal. State where Euclidean parallel uniqueness enters.Written reasoning: review required
• A transversal forms equal alternate interior angles with two distinct coplanar lines. Prove those lines parallel rather than assuming the converse of a theorem automatically holds.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• A transversal cuts two distinct coplanar lines with equal alternate interior angles of 64 degrees. Prove the lines parallel using the direct theorem and a constructed comparison line. A nondegenerate triangle has angles 64 and 49 degrees; derive its third angle and the exterior angle there from a parallel through that vertex.Written reasoning: review required

Track angles through a transversal — Euclidean and coordinate geometry foundations

• Prove that vertical angles are equal using straight-angle sums.Written reasoning: review required
• For two parallel lines cut by a transversal, prove same-side interior angles are supplementary using an alternate interior angle.Written reasoning: review required

Construct a point and justify why it works — Euclidean and coordinate geometry foundations

• Construct the perpendicular bisector of a segment AB and justify it, including the radius condition.Written reasoning: review required

Prove angle relations with parallel lines — Geometric definitions, motion witnesses and theorem proofs

• Derive the triangle angle sum by drawing a line through one vertex parallel to the opposite side. Then derive the exterior-angle theorem, explaining nondegeneracy.Written reasoning: review required

Prove midpoint and median structure — Geometric definitions, motion witnesses and theorem proofs

• Prove the segment joining the midpoints of AB and AC is parallel to BC and half its length, for any nondegenerate triangle.Written reasoning: review required
• Prove the three medians concur, not merely that two drawn segments appear to meet. Determine their division ratio from each vertex.Written reasoning: review required
• In nondegenerate ABC, AB=AC. Let M be the midpoint of BC. Prove the base angles are equal and AM is perpendicular to BC, with no assumption that the drawing is symmetric.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• For A=(−3,−1),B=(9,−1),C=(0,8), find all side midpoints and the centroid. Verify a midsegment’s direction and half-length, then prove the same centroid formula lies on all three medians for arbitrary nondegenerate vertices.Written reasoning: review required

Match triangles with sufficient evidence — Euclidean and coordinate geometry foundations

• If AB=AC, prove angles ABC and BCA are equal by comparing triangles ABC and ACB.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• In triangle ABC, D and E are the midpoints of AB and AC. Prove DE is parallel to BC and DE=BC/2 using dilation.Written reasoning: review required

Affine coordinates and barycentric reasoning — Axiomatic Euclidean and non-Euclidean geometry

• Prove the medians of a nondegenerate affine triangle meet at G=(A+B+C)/3 and divide each median in ratio 2:1.Written reasoning: review required

Characterize parallelograms and rectangles — Geometric definitions, motion witnesses and theorem proofs

• Prove that opposite sides and angles of a parallelogram are equal by drawing a diagonal. Do not assume those properties in the definition.Written reasoning: review required
• Prove both directions: a nondegenerate simple quadrilateral is a parallelogram iff its diagonals bisect each other.Written reasoning: review required
• Within a parallelogram ABCD, prove that equal diagonal lengths imply a rectangle. Show the converse too.Written reasoning: review required
• Show that equal diagonals alone do not make a quadrilateral a rectangle, using vertices (−4,0),(4,0),(2,3),(−2,3) in order.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• A quadrilateral has A=(0,0),B=(5,0),C=(7,4),D=(2,4). Prove it is a parallelogram by diagonal bisection. Decide whether it is a rectangle using its diagonals. State and prove the equal-diagonal rectangle criterion under the needed hypothesis.Written reasoning: review required

Choose coordinates that expose the geometry — Euclidean and coordinate geometry foundations

• Let A=(0,0), B=(p,0), D=(q,r), C=(p+q,r), with p>0 and r≠0. Prove the diagonals of this parallelogram bisect each other.Written reasoning: review required

Copy lengths and angles with a proof — Geometric constructions, circle proofs and coordinate loci

• Given a nonzero segment AB and a ray from P, give executable compass-and-straightedge steps to copy AB onto that ray. Explain why there is exactly one answer on the ray.Written reasoning: review required
• Copy a given angle between distinct nonopposite rays at A to a specified side of a ray from P. Use a common radius, a chord transfer and SSS; account for the two candidate circle intersections.Written reasoning: review required
• Construct the internal bisector of a nonstraight angle using two equal-radius arcs. Prove that the chosen ray is internal and bisects the angle.Written reasoning: review required
• Describe a paper-folding alternative to angle bisection and explain its exact mathematical idealization. Why is a crease that merely passes through the vertex insufficient?Written reasoning: review required

Construct perpendicular and parallel lines — Geometric constructions, circle proofs and coordinate loci

• Construct a perpendicular to line ℓ through a point P on ℓ. Give exact steps and a proof.Written reasoning: review required
• Given P outside line ℓ, construct its perpendicular foot using a circle that cuts ℓ twice. Explain how to choose a safe radius and why the resulting bisector passes through P.Written reasoning: review required
• Construct through an external point P a line parallel to ℓ by two methods: compass-and-straightedge perpendiculars, and ideal folds. Prove the result.Written reasoning: review required
• Prove that a point X is equidistant from distinct A,B if and only if X lies on the perpendicular bisector of AB. Include X at the midpoint.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• Given a segment UV and an angle between distinct nonopposite rays at W, copy the segment onto a supplied ray and copy the angle to a specified side of that ray. Then describe an ideal folding method to bisect the copied angle. Explain why copying a chord alone without transferring its radius does not fix the angle.Written reasoning: review required
• Given a line ℓ and external point P, construct a perpendicular and a parallel through P without measuring angles. Prove both. Also construct the midpoint of a separate segment of length 18 and explain why equal compass radii of 9 fail to determine its bisector line.Written reasoning: review required

Construct a point and justify why it works — Euclidean and coordinate geometry foundations

• Construct the perpendicular bisector of a segment AB and justify it, including the radius condition.Written reasoning: review required
• Construct the internal bisector of an angle strictly between 0° and 180° and prove it using congruent triangles.Written reasoning: review required
• Segment AB is 10 cm long. Equal-radius circles centered at A and B intersect twice only when their radius exceeds what value, in cm?Automatically checked response; reasoning still matters

Inscribe a square, hexagon and equilateral triangle — Geometric constructions, circle proofs and coordinate loci

• Give a compass-and-straightedge construction of a regular hexagon in a given circle with known center O and radius r>0. Prove the six steps close.Written reasoning: review required
• Construct an equilateral triangle inscribed in a given circle by using a regular hexagon. Prove that the selected vertices have equal pairwise distances.Written reasoning: review required
• Construct a square inscribed in a given circle. Explain why merely drawing two arbitrary diameters does not suffice.Written reasoning: review required
• A learner steps a compass opening of 0.99r around a circle of radius r and claims six steps still form an exact regular hexagon. Diagnose the claim without measuring the final gap.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• In a circle of radius 4, construct an inscribed regular hexagon, an equilateral triangle and a square. Prove regularity for each. Give the hexagon perimeter and the squared side lengths of the triangle and square.Written reasoning: review required

Construct a point and justify why it works — Euclidean and coordinate geometry foundations

• An equilateral triangle is constructed on a segment of length 6. What is the square of its altitude?Automatically checked response; reasoning still matters

Track a dilation beyond the origin — Geometric definitions, motion witnesses and theorem proofs

• Dilate P=(2,4) about O=(−1,2) by factor 3. Enter the image x-coordinate.Automatically checked response; reasoning still matters
• A segment has length 15/4. What is its image length under a positive dilation of factor 2/3?Automatically checked response; reasoning still matters
• Find the image of the full line y=2x+1 under dilation about (−1,2) by factor 3. Explain why substituting the new coordinates into the old equation would be wrong.Written reasoning: review required
• For factor k>0 and center O, prove that a line through O maps onto itself and a line not through O maps onto a parallel line. Include vertical lines without using slope.Written reasoning: review required
• A learner permits factor zero and claims the image of every segment is a parallel segment with equal endpoint angles. Diagnose the failure and explain why negative factors require a separately stated direction convention.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• Dilate about O=(2,−1) by factor 3/2. Find the image of y=−x+4 and of segment endpoints U=(0,1),V=(4,3). Verify its length scale. Explain how a positive dilation combined with motions yields the AA criterion rather than merely checking one drawing.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• Dilate point (5,3) by factor 2 about center (1,1). What is the image?Automatically checked response; reasoning still matters
• Similar triangles have corresponding sides 6 and 15. A second side of the smaller triangle is 8. Find the matching larger side.Automatically checked response; reasoning still matters
• In triangle ABC, D and E are the midpoints of AB and AC. Prove DE is parallel to BC and DE=BC/2 using dilation.Written reasoning: review required

Construct a similarity and derive AA — Geometric definitions, motion witnesses and theorem proofs

• Construct a similarity taking A=(0,0), B=(3,0), C=(0,4) to P=(2,−1), Q=(8,−1), R=(2,7). Verify all three vertices and explain what happens to sides and angles.Written reasoning: review required
• Two right triangles have legs (3,4) and (3,5). Does their common right angle prove similarity? Give a decisive test and explain why AA is stronger.Written reasoning: review required
• A=(0,0), B=(2,0), C=(0,1) and P=(4,2), Q=(0,2), R=(4,4) are proposed corresponding vertices. Find a similarity witness and discuss why a positive dilation and translations alone cannot suffice.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• Similar triangles have corresponding sides 6 and 15. A second side of the smaller triangle is 8. Find the matching larger side.Automatically checked response; reasoning still matters
• Two nondegenerate triangles have two corresponding angles equal. Which conclusion follows?Automatically checked response; reasoning still matters
• Derive why a triangle's area scales by k² under a positive dilation k, rather than k.Written reasoning: review required

Similarities and the hierarchy of invariants — Axiomatic Euclidean and non-Euclidean geometry

• Suppose d(Fx,Fy)=s d(x,y) for fixed s>0. Prove F(x)=sQx+b with Q orthogonal.Written reasoning: review required
• Prove a similarity preserves unsigned angles between nonzero vectors.Written reasoning: review required

Construct a similarity and derive AA — Geometric definitions, motion witnesses and theorem proofs

• Prove AA for existing nondegenerate triangles with two corresponding angles equal. Use only motions, positive dilation and uniqueness of a ray intersection.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• Dilate about O=(2,−1) by factor 3/2. Find the image of y=−x+4 and of segment endpoints U=(0,1),V=(4,3). Verify its length scale. Explain how a positive dilation combined with motions yields the AA criterion rather than merely checking one drawing.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• Two nondegenerate triangles have two corresponding angles equal. Which conclusion follows?Automatically checked response; reasoning still matters

Choose ratios relative to an angle — Euclidean and coordinate geometry foundations

• Explain why two right triangles with the same acute angle have equal opposite/hypotenuse ratios.Written reasoning: review required

Use proportional cuts and altitude similarity — Geometric definitions, motion witnesses and theorem proofs

• Prove that a line parallel to BC meeting the interiors of AB and AC at D,E cuts those sides proportionally, both AD/AB=AE/AC and AD/DB=AE/EC.Written reasoning: review required
• Suppose D,E lie inside AB,AC and AD/AB=AE/AC. Prove DE parallel to BC without citing the converse as an axiom.Written reasoning: review required
• Let ABC be right at C, and let CD be perpendicular to hypotenuse AB. Put AD=p, DB=q, AB=c, AC=b, BC=a. Derive b²=cp, a²=cq and then Pythagoras from AA.Written reasoning: review required
• A right triangle has hypotenuse 25 and one leg 15. Find its other leg, the two hypotenuse projections and its altitude. Explain why these formulas cannot be copied to an arbitrary oblique triangle.Written reasoning: review required

Construct witnesses and justify geometric conclusions — Geometric definitions, motion witnesses and theorem proofs

• In triangle XYZ, points D,E inside XY,XZ satisfy XY=21, XD=8, XZ=35 and XE=40/3. Prove DE parallel to YZ, with an auxiliary construction. Separately a right triangle has legs 7 and 24: derive its hypotenuse projections and altitude using AA, not a memorized area shortcut.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• In triangle ABC, D and E are the midpoints of AB and AC. Prove DE is parallel to BC and DE=BC/2 using dilation.Written reasoning: review required

Connect right angles with squared lengths — Euclidean and coordinate geometry foundations

• Prove a²+b²=c² using a square of side a+b tiled by four congruent right triangles with legs a,b and a central square of side c.Written reasoning: review required
• If a triangle has positive side lengths a,b,c with a²+b²=c², prove the angle opposite c is right by constructing a comparison triangle.Written reasoning: review required

Match triangles with sufficient evidence — Euclidean and coordinate geometry foundations

• Triangle ABC is congruent to triangle PQR in that order. BC=13/2 cm. Find QR in cm.Automatically checked response; reasoning still matters
• If AB=AC, prove angles ABC and BCA are equal by comparing triangles ABC and ACB.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• Similar triangles have corresponding sides 6 and 15. A second side of the smaller triangle is 8. Find the matching larger side.Automatically checked response; reasoning still matters
• In triangle ABC, D and E are the midpoints of AB and AC. Prove DE is parallel to BC and DE=BC/2 using dilation.Written reasoning: review required

Choose ratios relative to an angle — Euclidean and coordinate geometry foundations

• Explain why two right triangles with the same acute angle have equal opposite/hypotenuse ratios.Written reasoning: review required
• In a right triangle, the side opposite acute angle θ is 5 and the hypotenuse is 13. Find sin θ.Automatically checked response; reasoning still matters
• Derive sin²θ+cos²θ=1 for an acute angle in a right triangle.Written reasoning: review required

Choose ratios relative to an angle — Euclidean and coordinate geometry foundations

• If acute angles α and β are complementary in a right triangle, which identity holds?Automatically checked response; reasoning still matters

Choose ratios relative to an angle — Euclidean and coordinate geometry foundations

• A vertical mast stands on horizontal ground. From a ground-level observation point 24 m horizontally from its base, θ is the angle of elevation from the horizontal to the mast's top. If tan θ=7/24, find the mast height in metres.Automatically checked response; reasoning still matters

Connect right angles with squared lengths — Euclidean and coordinate geometry foundations

• A right triangle has legs 9 and 12. Find its hypotenuse.Automatically checked response; reasoning still matters
• A right triangle has hypotenuse 17 and one leg 8. Find the other leg.Automatically checked response; reasoning still matters

Derive triangle laws and measure inaccessible distances — Geometric constructions, circle proofs and coordinate loci

• Derive Δ=bc sin A/2 for 0<A<180 degrees. Draw or describe the altitude for acute and obtuse A rather than assuming its foot is always inside side c.Written reasoning: review required
• Two sides of lengths 8 and 9 include an angle of 30 degrees. Find the triangle area.Automatically checked response; reasoning still matters

Solve triangles from sides or an included angle — Trigonometry, polar geometry and precalculus representations

• Sides6 and8 enclose30°. Find the area.Automatically checked response; reasoning still matters

Derive triangle laws and measure inaccessible distances — Geometric constructions, circle proofs and coordinate loci

• Prove the sine law for every nondegenerate Euclidean triangle using the altitude area formula. Explain why it does not by itself resolve every SSA measurement.Written reasoning: review required
• Derive the cosine law for c opposite C using coordinates, including the obtuse case. Identify what a negative computed c² says about proposed data.Written reasoning: review required
• Synthetic survey data give A=30°, side a=6 m opposite A, and side b=10 m opposite B. Determine how many triangles are possible and approximate both B and C values to 0.1 degree. State a measurement that would choose one.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• Synthetic survey stations A,B are 80 m apart on level ground. A target T is located by angles TAB=50° and TBA=65° on the same side of AB. Find AT to 0.1 m. Separately two paths of lengths 9 m and 13 m meet at an included 120° angle; find endpoint separation exactly. Finally classify SSA data A=30°, a=7 m, b=12 m and approximate both possible B angles. State the needed physical assumptions.Written reasoning: review required

Solve triangles from sides or an included angle — Trigonometry, polar geometry and precalculus representations

• Place one side of length a along the x-axis and another of length b at included angle C. Derive c²=a²+b²−2ab cos C from endpoint distance.Written reasoning: review required
• Two sides of lengths5 and7 enclose60°. What is the square of the opposite side?Automatically checked response; reasoning still matters

Use the sine law without losing a triangle — Trigonometry, polar geometry and precalculus representations

• A triangle has A=30°, B=90° and side b=10 opposite B. Find side a.Automatically checked response; reasoning still matters
• For an acute given angle A and positive sides a,b, define h=b sin A. Explain the counts when a<h, a=h, h<a<b, and a≥b.Written reasoning: review required

Solve triangles from sides or an included angle — Trigonometry, polar geometry and precalculus representations

• Two sides of lengths5 and7 enclose60°. What is the square of the opposite side?Automatically checked response; reasoning still matters
• A learner enters sides2,3,5 into the cosine law and obtains an angle of180°. Explain why this is not a nondegenerate triangle, despite a computable cosine.Written reasoning: review required

Use the sine law without losing a triangle — Trigonometry, polar geometry and precalculus representations

• For A=30°, a=5 and b=8, how many nondegenerate triangles exist?Automatically checked response; reasoning still matters
• For A=120°, a=6 and b=4, explain why there is only one triangle even if a calculator suggests supplementary candidates. Give an exact expression for B.Written reasoning: review required

Prove circle similarity and recover a circle equation — Geometric constructions, circle proofs and coordinate loci

• Prove all positive-radius circles are similar. For centers C,D and radii r,s, give the point map and prove both inclusions of its image locus.Written reasoning: review required
• Map the circle centered at (−2,3) with radius 4 onto the circle centered at (5,−1) with radius 10. Write the map and verify a source point, then diagnose scaling about the origin alone.Written reasoning: review required
• Classify the real loci x²+y²+2x−10y+26=0 and x²+y²+2x−10y+30=0. A student calls both circles; explain the problem.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• Complete squares in x²+y²−2x+8y−8=0. Map this locus onto the radius-3 circle centered at (−2,5), proving onto equality. Then classify the equation obtained by replacing −8 with +17.Written reasoning: review required

Scale lengths, areas and arguments — Euclidean and coordinate geometry foundations

• Dilate point (5,3) by factor 2 about center (1,1). What is the image?Automatically checked response; reasoning still matters

Measure rotation and derive exact values — Trigonometry, polar geometry and precalculus representations

• An angle subtends a 15 cm arc on a circle of radius 6 cm. Give the angle in radians.Automatically checked response; reasoning still matters

Read the correct arcs in circle angle proofs — Geometric constructions, circle proofs and coordinate loci

• For an inscribed angle AXB, state exactly which arc AB is intercepted. If the minor arc AB measures 86 degrees, find the angle when X lies on the major arc and when X lies on the minor arc.Written reasoning: review required
• Prove the inscribed-angle theorem first when one side AX is a diameter, then explain how splitting or subtracting angles with diameter XO covers the other positions.Written reasoning: review required
• Prove the smaller angle APB between two tangents from external P equals 180 degrees minus the minor central angle AOB. Express it also using major and minor arc measures.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• Two tangents from P touch a circle at A,B and angle APB is 38 degrees. Find minor and major arc AB, then both possible inscribed angles AXB according to the arc containing X. Describe the diameter-circle tangent construction and prove it finds every contact.Written reasoning: review required

Read the arc before using the angle — Euclidean and coordinate geometry foundations

• An inscribed angle intercepts an arc of measure 146° that does not contain its vertex. Find the angle in degrees.Automatically checked response; reasoning still matters
• An inscribed angle subtending a diameter measures:Automatically checked response; reasoning still matters
• Prove that a line from the center perpendicular to a chord bisects that chord.Written reasoning: review required
• From an external point P, tangents PA and PB touch one circle. Prove PA=PB.Written reasoning: review required

Geometry foundations: calculation and proof assessment — Euclidean and coordinate geometry foundations

• A circle has center O and radius 9 cm. From point P with OP=15 cm a tangent touches at T. Find PT and justify the needed right angle. A separate chord AB subtends a central minor angle AOB of 110°. Find an inscribed angle ACB when C lies on the major arc AB, stating which arc it intercepts.Written reasoning: review required

Construct the circumcircle and incircle — Geometric constructions, circle proofs and coordinate loci

• Construct the circumcircle of an arbitrary nondegenerate triangle ABC. Prove the third perpendicular bisector passes through the constructed center, without assuming all centers are inside.Written reasoning: review required
• Prove that an interior point of an angle strictly between 0 and 180 degrees lies on the internal bisector iff its perpendicular distances to the two side lines are equal.Written reasoning: review required
• Construct an incircle of ABC and prove it is tangent to all three side segments, not just equidistant from two lines.Written reasoning: review required
• Derive Δ=rs for an incircle, defining every symbol. Diagnose the error in using a triangle vertex as the incircle center merely because it lies on two side lines.Written reasoning: review required

Read the correct arcs in circle angle proofs — Geometric constructions, circle proofs and coordinate loci

• Prove that opposite angles of a convex quadrilateral with four distinct vertices on a circle sum to 180 degrees. Why must the vertex order be stated?Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• For the triangle with vertices (0,0),(12,0),(0,5), construct its circumcircle and incircle. Find both centers and radii, then prove the incircle contacts the three side segments. A coordinate check may verify a construction but cannot replace its steps.Written reasoning: review required
• A convex quadrilateral ABCD has four distinct vertices in circular order. Prove both opposite-angle sums. If angle A=97 degrees and angle B=76 degrees, find C,D. Explain why replacing circular order by a crossed listing changes the statement.Written reasoning: review required

Geometry foundations: calculation and proof assessment — Euclidean and coordinate geometry foundations

• Points A=(0,0), B=(6,0) and C=(2,4) form a triangle. Describe a compass-and-straightedge construction of their circumcenter using perpendicular bisectors. Find its coordinates and radius, and prove the resulting circle passes through all three points. Explain why no circle can pass through three distinct collinear points.Written reasoning: review required

Locate triangle centers and test possible sides — Euclidean and coordinate geometry foundations

• If O lies on the perpendicular bisectors of AB and AC, prove OA=OB=OC and explain why this locates a circle through A,B,C.Written reasoning: review required
• Which lines meet at the circumcenter?Automatically checked response; reasoning still matters

Construct both tangents from an external point — Geometric constructions, circle proofs and coordinate loci

• Give executable compass-and-straightedge instructions for the tangents from P to a circle with center O and radius r, assuming OP>r>0. Prove correctness.Written reasoning: review required
• Show that the diameter-circle construction finds every possible tangent from P, not merely two candidate lines that work.Written reasoning: review required
• Classify tangent lines through P when OP>r, OP=r and OP<r for r>0. Explain how the auxiliary-circle construction degenerates at the boundary.Written reasoning: review required
• A learner draws a circle centered at P through O and connects its intersections with the original circle to P. Explain why this does not implement the diameter construction; give an exact counterexample.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• Two tangents from P touch a circle at A,B and angle APB is 38 degrees. Find minor and major arc AB, then both possible inscribed angles AXB according to the arc containing X. Describe the diameter-circle tangent construction and prove it finds every contact.Written reasoning: review required

Read the arc before using the angle — Euclidean and coordinate geometry foundations

• An inscribed angle subtending a diameter measures:Automatically checked response; reasoning still matters
• From external P, a tangent PT touches a circle centered O. OP=13 and radius OT=5. Find PT.Automatically checked response; reasoning still matters
• From an external point P, tangents PA and PB touch one circle. Prove PA=PB.Written reasoning: review required

Derive arc scaling and sector area — Geometric constructions, circle proofs and coordinate loci

• Prove that corresponding arcs under a dilation of positive factor k have lengths in ratio k. Do not assume the formula s=rθ to prove its own scaling premise.Written reasoning: review required
• Derive A=r²θ/2 for a circular sector with 0≤θ≤2π, using equal-angle partitions and a limiting extension to nonrational fractions of a turn.Written reasoning: review required
• A student uses A=r²·72/2 for a radius-5 sector with angle 72 degrees. Repair the calculation, give arc length and area exactly, and explain the units.Written reasoning: review required
• A curved strip is the difference of concentric sectors of radii 5 and 3 and central angle 2 radians. Find its area and the lengths of its two curved edges. Explain why the area is not width times outer arc length.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• A circular path has radius 9 m and a marked arc of length 12 m. Find its radian angle and sector area. Derive why doubling the radius at the same angle doubles arc length but quadruples area, starting with chord sums and sector partitions.Written reasoning: review required

Measure rotation and derive exact values — Trigonometry, polar geometry and precalculus representations

• An angle subtends a 15 cm arc on a circle of radius 6 cm. Give the angle in radians.Automatically checked response; reasoning still matters
• Write 225 degrees as kπ radians. Enter k exactly.Automatically checked response; reasoning still matters

Measure a region by small sectors — Calculus II: series, curves and applications

• The sector r=3, 0≤θ≤π/2 has area kπ. Find k.Automatically checked response; reasoning still matters
• Find the area between r=2 and r=1 over 0≤θ≤π/3.Written reasoning: review required

Prove circle similarity and recover a circle equation — Geometric constructions, circle proofs and coordinate loci

• Derive the equation of a circle centered at (h,k) with radius r>0 and prove that every real solution lies on it. Explain why squaring is reversible.Written reasoning: review required
• Complete squares in x²+y²−8x+6y−11=0. What is the squared radius?Automatically checked response; reasoning still matters
• Classify the real loci x²+y²+2x−10y+26=0 and x²+y²+2x−10y+30=0. A student calls both circles; explain the problem.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• Complete squares in x²+y²−2x+8y−8=0. Map this locus onto the radius-3 circle centered at (−2,5), proving onto equality. Then classify the equation obtained by replacing −8 with +17.Written reasoning: review required

Read conics from equations and geometric definitions — Trigonometry, polar geometry and precalculus representations

• For x²+y²−6x+4y−12=0, find the circle radius.Automatically checked response; reasoning still matters

Choose coordinates that expose the geometry — Euclidean and coordinate geometry foundations

• Find the distance from (−2,1) to (4,9).Automatically checked response; reasoning still matters

Read conics from equations and geometric definitions — Trigonometry, polar geometry and precalculus representations

• Derive the parabola with focus(2,0) and directrix x=−2 from equality of distances. Explain why squaring introduces no extraneous curve here.Written reasoning: review required
• For y²=12x, what is the x-coordinate of the focus?Automatically checked response; reasoning still matters

Derive the ellipse from its two foci — Geometric constructions, circle proofs and coordinate loci

• Derive x²/a²+y²/b²=1 from d₋+d₊=2a, where d₋=√((x+c)²+y²), d₊=√((x−c)²+y²), a>c>0 and b²=a²−c².Written reasoning: review required
• Prove the standard ellipse equation implies the original focal sum under a>c>0, b²=a²−c². Identify the sign check after squaring.Written reasoning: review required
• For distinct foci separated by 10, classify focal sums 8,10,14. Why does the middle case not allow division by positive b²?Written reasoning: review required
• Find the ellipse with foci (−1,3),(7,3) and sum 10. Verify (3,6) using the original distances.Written reasoning: review required

Derive both branches of a hyperbola — Geometric constructions, circle proofs and coordinate loci

• Derive x²/a²−y²/b²=1 from right-branch d₋−d₊=2a, with d₋=√((x+c)²+y²), d₊=√((x−c)²+y²), 0<a<c and b²=c²−a². Explain the left branch.Written reasoning: review required
• Prove that the full equation implies |d₋−d₊|=2a under 0<a<c,b²=c²−a². Include both branches and signs.Written reasoning: review required
• For foci (−3,0),(3,0), classify absolute differences 0,6,8 and compare with a difference strictly between 0 and 6.Written reasoning: review required
• Find the hyperbola with foci (1,−2),(11,−2) and absolute difference 6. Give vertices and show its asymptotes are not in the locus.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• A locus has foci (−2,1),(10,1) and distance sum 20. Derive its equation from the distance definition, prove the converse with signs, and verify the point (4,9) by distances. Explain the locus if the sum is instead 12.Written reasoning: review required
• A locus has foci (−2,−3),(8,−3) and absolute distance difference 8. Derive its two-branch equation and vertices. Prove the converse for both branches; then classify differences 0 and 11.Written reasoning: review required

Read conics from equations and geometric definitions — Trigonometry, polar geometry and precalculus representations

• For x²/25+y²/9=1, what is the full major-axis length?Automatically checked response; reasoning still matters
• Give a parametrization of(x−2)²/16+(y+1)²/4=1 tracing it once, and identify its centre and semiaxes.Written reasoning: review required
• Which equation describes a hyperbola rather than an ellipse?Automatically checked response; reasoning still matters

Conics as homogeneous quadratic geometry — Axiomatic Euclidean and non-Euclidean geometry

• Under x=My with invertible M, derive the matrix of the conic xᵀQx=0 in y-coordinates.Written reasoning: review required

Choose coordinates that expose the geometry — Euclidean and coordinate geometry foundations

• Let A=(0,0), B=(p,0), D=(q,r), C=(p+q,r), with p>0 and r≠0. Prove the diagonals of this parallelogram bisect each other.Written reasoning: review required
• For A=(0,0), B=(b,0), C=(u,v), b>0,v≠0, prove the segment joining the midpoints of AB and AC is parallel to BC and half as long without slopes.Written reasoning: review required

Prove and use coordinate line criteria — Geometric constructions, circle proofs and coordinate loci

• Prove that two distinct nonvertical lines are parallel iff their slopes agree. Explain why equal slopes alone do not imply distinct parallel lines.Written reasoning: review required
• Derive m₁m₂=−1 for perpendicular nonvertical lines using a quarter-turn of a direction vector. Handle the horizontal/vertical exception.Written reasoning: review required
• Through P=(4,−3), construct equations for the line parallel to x=−2 and the line perpendicular to x=−2. Explain their uniqueness.Written reasoning: review required
• A learner says lines with slopes 2 and −2 are perpendicular because the slopes have opposite signs. Refute the claim and give an actual perpendicular to y=2x+7 through (6,1).Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• A layout has A=(−1,−2), B=(7,4), D=(−4,2). Prove these define adjacent sides of a rectangle, construct C, and find area/perimeter. Locate P internally with AP:PB=3:1 and give an equation of the line through P perpendicular to AB. Describe a compass/parallel construction of that division.Written reasoning: review required

Choose coordinates that expose the geometry — Euclidean and coordinate geometry foundations

• What is the slope of the line through (3,−2) and (3,5)?Automatically checked response; reasoning still matters
• For A=(0,0), B=(b,0), C=(u,v), b>0,v≠0, prove the segment joining the midpoints of AB and AC is parallel to BC and half as long without slopes.Written reasoning: review required

Measure vector alignment and projection — Trigonometry, polar geometry and precalculus representations

• Which vector is perpendicular to(2,−3)?Automatically checked response; reasoning still matters
• Find the angle between nonzero vectors u=(1,0) and v=(1,1). Explain the role of normalization in the angle formula.Written reasoning: review required

Divide a directed segment in a chosen ratio — Geometric constructions, circle proofs and coordinate loci

• Derive P=(nA+mB)/(m+n) for AP:PB=m:n internally. State all positivity and ordering assumptions.Written reasoning: review required
• Using unmarked straightedge and compass, divide a given nonzero segment AB internally in the ratio 3:2. Prove the construction.Written reasoning: review required
• For A=(1,−2), B=(5,4), interpret P=A+(3/2)(B−A). Find P and unsigned AP:PB. Why does the internal section formula fail for that ratio?Written reasoning: review required
• For A=(−4,0), B=(8,6), P has AP:PB=2:1 internally. Find P, express it from B toward A, and diagnose using t=2/1.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• A layout has A=(−1,−2), B=(7,4), D=(−4,2). Prove these define adjacent sides of a rectangle, construct C, and find area/perimeter. Locate P internally with AP:PB=3:1 and give an equation of the line through P perpendicular to AB. Describe a compass/parallel construction of that division.Written reasoning: review required

Choose coordinates that expose the geometry — Euclidean and coordinate geometry foundations

• Find the x coordinate of the midpoint of (−4,7) and (10,−3).Automatically checked response; reasoning still matters

Betweenness, separation and Pasch — Axiomatic Euclidean and non-Euclidean geometry

• On an ordered coordinate line A=−4 and C=8, B=(1−t)A+tC with t=1/4. Find B.Automatically checked response; reasoning still matters

Affine coordinates and barycentric reasoning — Axiomatic Euclidean and non-Euclidean geometry

• Prove an affine map preserves a directed division parameter t in P+t(Q−P), and explain why it need not preserve lengths.Written reasoning: review required

Measure coordinate polygons with justified lengths — Geometric constructions, circle proofs and coordinate loci

• Rectangle vertices in order are A=(1,1), B=(4,5), C=(−8,14), D=(−11,10). Find its perimeter.Automatically checked response; reasoning still matters
• Find the area of the triangle with vertices (0,0),(4,2),(−2,6).Automatically checked response; reasoning still matters
• For A=(−2,1), B=(1,5), D=(−14,10), construct the fourth vertex C of a rectangle, then find area and perimeter with justification.Written reasoning: review required
• Derive the area |ps−qr|/2 of a triangle with side vectors u=(p,q),v=(r,s) from one vertex. Do not merely quote the formula.Written reasoning: review required
• A simple polygon has ordered vertices (0,0),(6,0),(6,4),(3,2),(0,4). Find its area by rectangle decomposition and identify its concave vertex.Written reasoning: review required
• A learner connects (0,0),(4,3),(4,0),(0,3) in that order and calls it a rectangle of area 12. Diagnose and supply a valid order and perimeter.Written reasoning: review required

Construct, prove and measure unfamiliar configurations — Geometric constructions, circle proofs and coordinate loci

• A layout has A=(−1,−2), B=(7,4), D=(−4,2). Prove these define adjacent sides of a rectangle, construct C, and find area/perimeter. Locate P internally with AP:PB=3:1 and give an equation of the line through P perpendicular to AB. Describe a compass/parallel construction of that division.Written reasoning: review required

Choose coordinates that expose the geometry — Euclidean and coordinate geometry foundations

• Find the distance from (−2,1) to (4,9).Automatically checked response; reasoning still matters

Decompose a region without counting it twice — Euclidean and coordinate geometry foundations

• A trapezoid has parallel bases 7 and 13 cm and height 4 cm. Find its area in cm².Automatically checked response; reasoning still matters
• Derive the trapezoid area formula for bases a,b and perpendicular height h by making two copies.Written reasoning: review required

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• A radius-2 circle has an inscribed regular 12-gon of side √6−√2 and apothem (√6+√2)/2. Derive its perimeter and area, derive the corresponding tangent polygon perimeter, and bracket circle area. What further argument gives exact area πr²?Written reasoning: review required
• A right prism has a right-triangle base with legs 3 cm and 8 cm and height 7 cm. An oblique prism has rectangular base 3 cm by 4 cm and perpendicular height 7 cm. Compare volumes using complete section rules. Why would agreement at only three heights be insufficient?Written reasoning: review required
• A square pyramid has base side 5 cm and height 12 cm. A right cone has radius 5 cm and slant length 13 cm. Find both volumes. For the pyramid use stack gap Bh/n to choose an integer n guaranteeing a gap smaller than 1/2 cm³, and explain the limiting one-third factor.Written reasoning: review required

Explain circle formulas by polygon bounds — Sections, solid measures and geometric design

• Using corresponding inscribed and circumscribed polygons, explain why scaling a circle by factor k>0 scales its limiting circumference by k. Deduce why C/(2r) is independent of r.Written reasoning: review required
• Let an inscribed regular polygon have perimeter p and apothem a in a circle of radius r. Its corresponding tangent polygon has perimeter q=pr/a. Use their triangle decompositions to explain the limiting area πr².Written reasoning: review required
• A learner replaces a circle by a regular 100-gon and says its perimeter is exactly 2πr. Explain the error and describe what additional bounds make an approximation trustworthy.Written reasoning: review required

Compare volumes through complete section rules — Sections, solid measures and geometric design

• Explain why a rectangular box with rational side lengths 3/2,5/3 and 2 has volume 5 cubic units by subdividing into equal cubes.Written reasoning: review required
• Compare a right prism and an oblique prism with the same polygonal base area B and the same perpendicular height h. Explain why their volumes agree even if corresponding sections are horizontally displaced.Written reasoning: review required

Derive the one-third volume factor — Sections, solid measures and geometric design

• Given Σ(k=1..n)k²=n(n+1)(2n+1)/6, evaluate the limit of (1/n)Σ(k=1..n)(k/n)² using ordinary algebra, and connect it to pyramid volume.Written reasoning: review required
• An oblique pyramid and a right pyramid have the same base area and perpendicular height. Use sections, not appearance, to justify equal volumes.Written reasoning: review required

Build volume from cross-sectional area — Integral calculus: accumulation, methods and applications

• The region 0≤y≤x, 0≤x≤2 rotates about the x-axis. Write its volume as kπ and find k.Automatically checked response; reasoning still matters
• A solid has square cross sections perpendicular to x, each side 2−x for 0≤x≤2. Find its volume.Automatically checked response; reasoning still matters

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• A right prism has a right-triangle base with legs 3 cm and 8 cm and height 7 cm. An oblique prism has rectangular base 3 cm by 4 cm and perpendicular height 7 cm. Compare volumes using complete section rules. Why would agreement at only three heights be insufficient?Written reasoning: review required
• For a sphere of radius 6 cm, construct the cylinder-minus-double-cone comparison. At central height z=2 cm calculate both section areas, then prove equality at every height and compute sphere volume. Why is one matching height insufficient?Written reasoning: review required

Compare volumes through complete section rules — Sections, solid measures and geometric design

• Compare a right prism and an oblique prism with the same polygonal base area B and the same perpendicular height h. Explain why their volumes agree even if corresponding sections are horizontally displaced.Written reasoning: review required
• A cylinder and another solid have equal section areas at heights 0,1 and 2. Is that enough to conclude equal volumes? Supply a counterexample section rule on 0≤z≤2 and explain the missing requirement.Written reasoning: review required

Recover sphere volume from a comparison solid — Sections, solid measures and geometric design

• For a sphere of radius R, describe the cylinder and removed cones that have matching horizontal section areas. Verify the equality for every height z between −R and R.Written reasoning: review required
• Derive the sphere volume from that comparison using established cylinder and cone volumes. State the comparison principle needed.Written reasoning: review required

Triple integrals in Cartesian, cylindrical and spherical coordinates — Multivariable calculus: geometry, change and accumulation

• Find the coefficient of π in the volume of a ball of radius 3.Automatically checked response; reasoning still matters
• Set up and evaluate the volume of x,y,z≥0, x+y+z≤2.Written reasoning: review required

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• A square pyramid has base side 5 cm and height 12 cm. A right cone has radius 5 cm and slant length 13 cm. Find both volumes. For the pyramid use stack gap Bh/n to choose an integer n guaranteeing a gap smaller than 1/2 cm³, and explain the limiting one-third factor.Written reasoning: review required
• For a sphere of radius 6 cm, construct the cylinder-minus-double-cone comparison. At central height z=2 cm calculate both section areas, then prove equality at every height and compute sphere volume. Why is one matching height insufficient?Written reasoning: review required
• Synthetic measurements give object radii 2 cm at heights 0 and 3 cm, and radius 0 at height 6 cm. Compare a radius-2,height-6 cylinder with a radius-2,height-3 cylinder joined to a radius-2,height-3 cone. Displacement is 50±2 cm³. Use 3.14159<π<3.14160 to assess both models and state needed further evidence.Written reasoning: review required

Choose and test a model for a measured object — Sections, solid measures and geometric design

• A square-based decorative pyramid has base side 12 cm and perpendicular height 5 cm. Find its external volume in cm³.Automatically checked response; reasoning still matters
• Synthetic radius measurements of a solid at heights 0,2,4,6 cm are 3,3,2,0 cm. Compare an ideal radius 3,height 6 cylinder with a cylinder of height 2 joined to a cone of height 4, both radius 3. Which better respects the measured taper, and what remains uncertain?Written reasoning: review required
• A synthetic sphere model has radius 3 cm and predicts 36πcm³. A displacement reading is 114±2 cm³, interpreted as inclusive bounds. Using 3.14159<π<3.14160, decide whether the prediction lies in the interval and what that does not prove.Written reasoning: review required
• A cylindrical container has external radius 5 cm and internal radius 4.8 cm, height 10 cm, with no bottom or lid in this idealized model. Determine material volume and explain whether external capacity is the right quantity.Written reasoning: review required
• A cone radius is bounded by[1.9,2.1]cm and height by[5.8,6.2]cm. Give exact volume bounds as multiples of π and explain why a midpoint calculation alone is insufficient.Written reasoning: review required

Derive the one-third volume factor — Sections, solid measures and geometric design

• A right cone has radius 6 cm and slant length 10 cm. Find its volume and explain why slant length cannot replace perpendicular height.Written reasoning: review required

Relate simultaneous rates before substituting — Differential calculus: techniques and applications

• A cone-shaped tank has height 6 m and radius 3 m. Water enters at 2 m³/min. Derive the height rate when the water depth is 2 m.Written reasoning: review required

Triple integrals in Cartesian, cylindrical and spherical coordinates — Multivariable calculus: geometry, change and accumulation

• Find the coefficient of π in the volume of a ball of radius 3.Automatically checked response; reasoning still matters
• Set up and evaluate the volume of x,y,z≥0, x+y+z≤2.Written reasoning: review required

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• In the cube 0≤x,y,z≤3, describe the section x+y+z=2 and prove its shape. Then describe the section in the vertical plane x=y. Explain why a perspective outline alone would not settle either answer.Written reasoning: review required
• Rotate the filled rectangle 2≤x≤5,0≤z≤3 first about the z-axis and then about the x-axis. Describe sections, dimensions and exact volumes. Does rotating only its boundary fill the same interiors?Written reasoning: review required

Read a solid through its sections — Sections, solid measures and geometric design

• Describe the section of a cube of side 4 by the vertical plane containing two opposite vertical edges. Give side lengths and justify the shape.Written reasoning: review required
• In the cube 0≤x,y,z≤2, determine the section x+y+z=1. Give its vertices and prove the shape.Written reasoning: review required
• Describe a horizontal and an oblique section of a right circular cylinder. State conditions under which the oblique section is an ellipse rather than a truncated region touching an end cap.Written reasoning: review required

Predict what a rotating region fills — Sections, solid measures and geometric design

• For the rectangle 0≤x≤3,0≤z≤5, compare rotation about the z-axis with rotation about the x-axis. Describe both solids and give the ratio of the second volume to the first.Written reasoning: review required
• Rotate the filled right triangle with vertices (0,0),(4,0),(0,6), using horizontal x and vertical z, about the z-axis. Derive the radius of the horizontal section at height z and identify the solid.Written reasoning: review required
• Rotate the filled right half-disk x≥0,x²+z²≤9 about the z-axis. Compare rotating only its semicircular arc.Written reasoning: review required
• A full disk x²+z²≤4 is rotated about its vertical diameter. A learner adds the swept volumes of its left and right halves. Diagnose the error and give the correct set description.Written reasoning: review required

Build volume from cross-sectional area — Integral calculus: accumulation, methods and applications

• Rotate the region between y=x and y=0 for 0≤x≤1 about y=−1. Set up and evaluate the volume.Written reasoning: review required
• A solid has equilateral triangular sections perpendicular to x, each side x for 0≤x≤3. Derive its volume.Written reasoning: review required

Count thin cylindrical layers — Integral calculus: accumulation, methods and applications

• Why does 2π∫₋₁¹ |x| dx double-count the solid formed by rotating the rectangle −1≤x≤1, 0≤y≤1 about the y-axis?Written reasoning: review required

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• Synthetic measurements give object radii 2 cm at heights 0 and 3 cm, and radius 0 at height 6 cm. Compare a radius-2,height-6 cylinder with a radius-2,height-3 cylinder joined to a radius-2,height-3 cone. Displacement is 50±2 cm³. Use 3.14159<π<3.14160 to assess both models and state needed further evidence.Written reasoning: review required

Choose and test a model for a measured object — Sections, solid measures and geometric design

• Synthetic radius measurements of a solid at heights 0,2,4,6 cm are 3,3,2,0 cm. Compare an ideal radius 3,height 6 cylinder with a cylinder of height 2 joined to a cone of height 4, both radius 3. Which better respects the measured taper, and what remains uncertain?Written reasoning: review required
• A synthetic sphere model has radius 3 cm and predicts 36πcm³. A displacement reading is 114±2 cm³, interpreted as inclusive bounds. Using 3.14159<π<3.14160, decide whether the prediction lies in the interval and what that does not prove.Written reasoning: review required
• A cylindrical container has external radius 5 cm and internal radius 4.8 cm, height 10 cm, with no bottom or lid in this idealized model. Determine material volume and explain whether external capacity is the right quantity.Written reasoning: review required
• A cone radius is bounded by[1.9,2.1]cm and height by[5.8,6.2]cm. Give exact volume bounds as multiples of π and explain why a midpoint calculation alone is insufficient.Written reasoning: review required

Choose ratios relative to an angle — Euclidean and coordinate geometry foundations

• A vertical mast stands on horizontal ground. From a ground-level observation point 24 m horizontally from its base, θ is the angle of elevation from the horizontal to the mast's top. If tan θ=7/24, find the mast height in metres.Automatically checked response; reasoning still matters

Relate simultaneous rates before substituting — Differential calculus: techniques and applications

• A cone-shaped tank has height 6 m and radius 3 m. Water enters at 2 m³/min. Derive the height rate when the water depth is 2 m.Written reasoning: review required

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• Compare closed boxes A (6×10×20 cm) and B (10×10×12 cm), with last dimension designated height. Require capacity ≥ 1200 cm³, height ≤ 20 cm and area ≤ 760 cm². Sheet area density is 0.15 g/cm². Calculate volumes, areas and ideal masses; choose a feasible lighter design, then reassess with height ≤ 15 cm. Could volume density replace area density without more data?Written reasoning: review required

Make density and design decisions from geometry — Sections, solid measures and geometric design

• A synthetic district has 3600 residents over 2.4 km². What is its mean population density in residents/km²?Automatically checked response; reasoning still matters
• A uniform rectangular block has volume 80 cm³ and density 27/10 g/cm³. Find its mass in grams.Automatically checked response; reasoning still matters
• A district has 3600 residents, total area 2.4 km² and uninhabited water area 0.6 km². Compare total-area and land-area population densities. Explain which denominator is appropriate.Written reasoning: review required
• A closed box has surface area 680 cm² and a sheet material has area density 0.12 g/cm². Find ideal sheet mass. Explain why multiplying by a volume density instead would require more information.Written reasoning: review required
• Two equal-volume regions of one object have densities 2 and 5 g/cm³. Is multiplying the entire volume by 2 or 5 valid? Derive the correct average and explain what changes for unequal region volumes.Written reasoning: review required

Double integrals, accumulation and density — Multivariable calculus: geometry, change and accumulation

• A lamina on 0≤x≤2, 0≤y≤1 has density 3x. Find its mass.Automatically checked response; reasoning still matters
• For density ρ=x+1 on the unit square, compute mass and x-coordinate of the center of mass.Written reasoning: review required

Parametric surfaces and scalar surface integrals — Multivariable calculus: geometry, change and accumulation

• A surface patch has area 7 and constant surface density 3. Find its mass.Automatically checked response; reasoning still matters

Solid geometry: sections, limits and design evidence — Sections, solid measures and geometric design

• Compare closed boxes A (6×10×20 cm) and B (10×10×12 cm), with last dimension designated height. Require capacity ≥ 1200 cm³, height ≤ 20 cm and area ≤ 760 cm². Sheet area density is 0.15 g/cm². Calculate volumes, areas and ideal masses; choose a feasible lighter design, then reassess with height ≤ 15 cm. Could volume density replace area density without more data?Written reasoning: review required

Make density and design decisions from geometry — Sections, solid measures and geometric design

• Compare closed rectangular boxes A (8×10×15 cm) and B (10×10×12 cm). Require capacity at least 1200 cm³ and height at most 16 cm, with the last listed dimension designated height. Compute capacities and areas, choose between the feasible designs to minimize sheet area, then explain what changes if the height limit tightens to 14 cm.Written reasoning: review required
• A closed box has surface area 680 cm² and a sheet material has area density 0.12 g/cm². Find ideal sheet mass. Explain why multiplying by a volume density instead would require more information.Written reasoning: review required

Combine meaningful functions into a quadratic decision — Functions in graphs, measurements and growth decisions

• A proper rectangle has perimeter 28 m and one side x m. Construct its area function and graph description, using factored and vertex forms. State the contextual domain, zero-area interpretation and maximum.Written reasoning: review required

Compare all candidates before choosing a design — Differential calculus: techniques and applications

• A rectangle uses 60 m of fencing on three sides, with the fourth side along a wall. Find the perpendicular width that maximizes area.Automatically checked response; reasoning still matters
• Cut equal squares of side x from each corner of a 12 by 12 sheet and fold an open box. Find the volume-maximizing x and justify the feasible domain and global result.Written reasoning: review required