Construct, prove and measure unfamiliar configurations
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 120 minutes, as an unvalidated planning estimate. This collection contains 12 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Give executable permitted-tool steps for constructions, not only final lengths or sketches. State intersection choices and degenerate cases.
Show original unsquared locus conditions when deriving equations. Calculator measurements require stated units, mode and sensible rounding.
These public mixed tasks require mathematical review; automatic completion is not a certification of construction or proof proficiency.
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0 of 12 tasks have recorded responses. No overall mastery score is assigned.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
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.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
