Construct witnesses and justify geometric conclusions
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 minutes, as an unvalidated planning estimate. This collection contains 10 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 labeled constructions or coordinate rules and justify every implication; a plausible sketch alone is not a proof.
State nondegeneracy, correspondence and Euclidean assumptions where needed. These public written tasks require mathematical review, not automatic mastery credit.
Cases are independent of the worked examples. You may consult definitions, but explain the mechanism in your own reasoning.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
A proposed circle consists of every point at distance at most 4 from (−1,3). Correct the definition and classify (3,3),(−1,3),(−1,−1). Then explain why two disjoint finite segments need not lie on parallel lines, giving coordinates.
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.
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.
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 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.
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 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.
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.
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.
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 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.
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.
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.
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 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.
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 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.
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 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.
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
A proposed circle consists of every point at distance at most 4 from (−1,3). Correct the definition and classify (3,3),(−1,3),(−1,−1). Then explain why two disjoint finite segments need not lie on parallel lines, giving coordinates.
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.
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.
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.
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.
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.
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.
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.
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.
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.
