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Transformations and Congruence Unit Test
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Below is a ready-to-print unit test on Transformations and Congruence (with directions, 10 questions, point values) plus a complete answer key and scoring rubric. You can copy, edit, or print as needed.
Transformations and Congruence — Unit Test
Time: 50 minutes
Show all work. Use correct notation for transformations (T for translation, R for rotation, r for reflection, D for dilation). Calculators allowed for arithmetic only.
Total: 50 points
Questions
1. (3 points)
Which single transformation always reverses orientation of a figure (changes clockwise ordering to counterclockwise ordering)?
A. Rotation
B. Translation
C. Reflection
D. Dilation (with positive scale factor)
Circle the correct answer and briefly justify in one sentence.
2. (4 points)
Triangle ABC has vertices A(1, 2), B(3, 4), C(0, 1). Find the coordinates of A', B', C' after the translation T(x, y) = (x + 3, y − 2). Show your work.
3. (3 points)
Rotate the point P(4, −1) 90° counterclockwise about the origin. Give the coordinates of the image.
4. (2 points)
Reflect the point Q(2, 5) across the line y = x. Give the image coordinates.
5. (3 points)
A dilation D centered at the origin with scale factor k maps R(1, −3) to R'. If k = −2, find R'. Explain briefly what a negative scale factor does to the location of a point.
6. (4 points)
Two triangles have side lengths 5, 7, 8 and 7, 8, 5 (corresponding sides listed in some order). State which congruence criterion you would use to prove the triangles are congruent, name the criterion (SSS, SAS, ASA, AAS, or HL), and give a one-sentence justification.
7. (4 points)
Apply the following composition of transformations to the point (3, 0): first rotate 90° counterclockwise about the origin, then translate by (2, −1). Give the final coordinates and show steps.
8. (4 points)
Find the center and scale factor of the dilation that maps P(2, 3) to P'(6, 9). Show how you determined both the center and k.
9. (8 points)
Triangle ABC has A(2, 3), B(5, 3), C(2, 6). Triangle A'B'C' has A'(−2, −2), B'(−2, 1), C'(1, −2).
a) (4 points) Give a sequence of rigid transformations (reflections, rotations, translations) that maps triangle ABC onto triangle A'B'C'. Be explicit (name lines/angles/translation vectors).
b) (4 points) Using your sequence, explain why triangles ABC and A'B'C' are congruent.
10. (5 points — Challenge)
Determine the single rigid transformation (type and angle) equivalent to the composition: reflect across the line y = x, then reflect across the x-axis. Describe the equivalent transformation algebraically (a rule for (x, y)) and name the transformation (e.g., rotation by ___ degrees about the origin).
Answer Key and Scoring Rubric
1. (3 points)
Answer: C — Reflection.
Reason: Reflection reverses orientation. (1 point for correct choice, 2 points for brief justification)
2. (4 points)
Compute:
A' = (1 + 3, 2 − 2) = (4, 0)
B' = (3 + 3, 4 − 2) = (6, 2)
C' = (0 + 3, 1 − 2) = (3, −1)
(1 point each coordinate pair, 1 point for showing translation comprehension)
3. (3 points)
Rule for 90° CCW about origin: (x, y) → (−y, x).
P(4, −1) → P' = (−(−1), 4) = (1, 4).
(3 points total)
4. (2 points)
Reflection across y = x swaps coordinates: (x, y) → (y, x).
Q(2, 5) → Q' = (5, 2).
(2 points)
5. (3 points)
R(1, −3), k = −2:
R' = (k·x, k·y) = (−2·1, −2·(−3)) = (−2, 6).
Brief explanation: A negative scale factor scales distances by |k| and also maps points through the center to the opposite side (a 180° rotation about the center combined with dilation). (2 points for coordinates, 1 point for explanation)
6. (4 points)
Criterion: SSS (Side-Side-Side).
Justification: All three pairs of corresponding side lengths are equal (5, 7, 8 correspond to 5, 7, 8), so triangles are congruent by SSS. (4 points: 2 for naming, 2 for justification)
7. (4 points)
Step 1: Rotate (3, 0) 90° CCW → (0, 3).
Step 2: Translate by (2, −1): (0 + 2, 3 − 1) = (2, 2).
Final answer: (2, 2). (2 points for rotation step, 2 points for translation/computation)
8. (4 points)
We see P(2, 3) → P'(6, 9). The vector from the origin: (2, 3) scaled by 3 gives (6, 9). So the dilation is centered at the origin and k = 3.
(If student shows the calculation k = 6/2 = 3 and verifies 9/3 = 3, award full credit.) (4 points: 2 for center, 2 for scale factor and justification)
9. (8 points)
a) (4 points) Sequence:
Reflect ABC across the line y = x, producing points:
A → (3, 2), B → (3, 5), C → (6, 2).
Then translate by vector (−5, −4) (T(x, y) = (x − 5, y − 4)):
A' = (3 − 5, 2 − 4) = (−2, −2)
B' = (3 − 5, 5 − 4) = (−2, 1)
C' = (6 − 5, 2 − 4) = (1, −2)
So the sequence is: r(y = x) then T(−5, −4).
Give full credit if student lists the same or any equivalent sequence of rigid motions (e.g., translate then reflect appropriately), with correct vectors/targets.
b) (4 points) Explanation:
Reflections and translations are rigid motions (isometries) — they preserve distances and angles. Since ABC is carried exactly onto A'B'C' by a composition of rigid motions, the triangles are congruent. (4 points: 2 for identification of rigid motions preserving congruence, 2 for concluding congruence)
10. (5 points)
Compute composition on a general point (x, y):
First reflect across y = x → (y, x).
Then reflect across the x-axis → (y, −x).
So the overall map is (x, y) → (y, −x).
Interpretation: (x, y) → (y, −x) is a rotation of 90° clockwise about the origin (equivalently a rotation of 270° CCW).
Algebraic rule: (x, y) → (y, −x). Name: rotation 90° clockwise about the origin.
(5 points: 2 for algebraic rule, 3 for correct identification of rotation and angle)
Total points check: 50.
Notes for grading and variations
- For questions with multiple-step coordinate work, award partial credit for correct intermediate steps.
- For Q9 a student could present an equivalent sequence (for example, translate first and then reflect across appropriately shifted line). Accept any correct sequence of rigid motions that produces the correct images; require explanation showing each image computed.
- If you prefer more multiple-choice items or diagram-based problems, tell me the grade level and I can produce a variant with figures instead of coordinate-only questions.
If you want, I can:
- Provide a printable PDF version.
- Replace any question with a diagram-based question (I can describe or give coordinates for drawing).
- Adjust difficulty for middle school, high-school geometry, or honors/IB level.
Transformations and Congruence — Unit Test
Time: 50 minutes
Show all work. Use correct notation for transformations (T for translation, R for rotation, r for reflection, D for dilation). Calculators allowed for arithmetic only.
Total: 50 points
Questions
1. (3 points)
Which single transformation always reverses orientation of a figure (changes clockwise ordering to counterclockwise ordering)?
A. Rotation
B. Translation
C. Reflection
D. Dilation (with positive scale factor)
Circle the correct answer and briefly justify in one sentence.
2. (4 points)
Triangle ABC has vertices A(1, 2), B(3, 4), C(0, 1). Find the coordinates of A', B', C' after the translation T(x, y) = (x + 3, y − 2). Show your work.
3. (3 points)
Rotate the point P(4, −1) 90° counterclockwise about the origin. Give the coordinates of the image.
4. (2 points)
Reflect the point Q(2, 5) across the line y = x. Give the image coordinates.
5. (3 points)
A dilation D centered at the origin with scale factor k maps R(1, −3) to R'. If k = −2, find R'. Explain briefly what a negative scale factor does to the location of a point.
6. (4 points)
Two triangles have side lengths 5, 7, 8 and 7, 8, 5 (corresponding sides listed in some order). State which congruence criterion you would use to prove the triangles are congruent, name the criterion (SSS, SAS, ASA, AAS, or HL), and give a one-sentence justification.
7. (4 points)
Apply the following composition of transformations to the point (3, 0): first rotate 90° counterclockwise about the origin, then translate by (2, −1). Give the final coordinates and show steps.
8. (4 points)
Find the center and scale factor of the dilation that maps P(2, 3) to P'(6, 9). Show how you determined both the center and k.
9. (8 points)
Triangle ABC has A(2, 3), B(5, 3), C(2, 6). Triangle A'B'C' has A'(−2, −2), B'(−2, 1), C'(1, −2).
a) (4 points) Give a sequence of rigid transformations (reflections, rotations, translations) that maps triangle ABC onto triangle A'B'C'. Be explicit (name lines/angles/translation vectors).
b) (4 points) Using your sequence, explain why triangles ABC and A'B'C' are congruent.
10. (5 points — Challenge)
Determine the single rigid transformation (type and angle) equivalent to the composition: reflect across the line y = x, then reflect across the x-axis. Describe the equivalent transformation algebraically (a rule for (x, y)) and name the transformation (e.g., rotation by ___ degrees about the origin).
Answer Key and Scoring Rubric
1. (3 points)
Answer: C — Reflection.
Reason: Reflection reverses orientation. (1 point for correct choice, 2 points for brief justification)
2. (4 points)
Compute:
A' = (1 + 3, 2 − 2) = (4, 0)
B' = (3 + 3, 4 − 2) = (6, 2)
C' = (0 + 3, 1 − 2) = (3, −1)
(1 point each coordinate pair, 1 point for showing translation comprehension)
3. (3 points)
Rule for 90° CCW about origin: (x, y) → (−y, x).
P(4, −1) → P' = (−(−1), 4) = (1, 4).
(3 points total)
4. (2 points)
Reflection across y = x swaps coordinates: (x, y) → (y, x).
Q(2, 5) → Q' = (5, 2).
(2 points)
5. (3 points)
R(1, −3), k = −2:
R' = (k·x, k·y) = (−2·1, −2·(−3)) = (−2, 6).
Brief explanation: A negative scale factor scales distances by |k| and also maps points through the center to the opposite side (a 180° rotation about the center combined with dilation). (2 points for coordinates, 1 point for explanation)
6. (4 points)
Criterion: SSS (Side-Side-Side).
Justification: All three pairs of corresponding side lengths are equal (5, 7, 8 correspond to 5, 7, 8), so triangles are congruent by SSS. (4 points: 2 for naming, 2 for justification)
7. (4 points)
Step 1: Rotate (3, 0) 90° CCW → (0, 3).
Step 2: Translate by (2, −1): (0 + 2, 3 − 1) = (2, 2).
Final answer: (2, 2). (2 points for rotation step, 2 points for translation/computation)
8. (4 points)
We see P(2, 3) → P'(6, 9). The vector from the origin: (2, 3) scaled by 3 gives (6, 9). So the dilation is centered at the origin and k = 3.
(If student shows the calculation k = 6/2 = 3 and verifies 9/3 = 3, award full credit.) (4 points: 2 for center, 2 for scale factor and justification)
9. (8 points)
a) (4 points) Sequence:
Reflect ABC across the line y = x, producing points:
A → (3, 2), B → (3, 5), C → (6, 2).
Then translate by vector (−5, −4) (T(x, y) = (x − 5, y − 4)):
A' = (3 − 5, 2 − 4) = (−2, −2)
B' = (3 − 5, 5 − 4) = (−2, 1)
C' = (6 − 5, 2 − 4) = (1, −2)
So the sequence is: r(y = x) then T(−5, −4).
Give full credit if student lists the same or any equivalent sequence of rigid motions (e.g., translate then reflect appropriately), with correct vectors/targets.
b) (4 points) Explanation:
Reflections and translations are rigid motions (isometries) — they preserve distances and angles. Since ABC is carried exactly onto A'B'C' by a composition of rigid motions, the triangles are congruent. (4 points: 2 for identification of rigid motions preserving congruence, 2 for concluding congruence)
10. (5 points)
Compute composition on a general point (x, y):
First reflect across y = x → (y, x).
Then reflect across the x-axis → (y, −x).
So the overall map is (x, y) → (y, −x).
Interpretation: (x, y) → (y, −x) is a rotation of 90° clockwise about the origin (equivalently a rotation of 270° CCW).
Algebraic rule: (x, y) → (y, −x). Name: rotation 90° clockwise about the origin.
(5 points: 2 for algebraic rule, 3 for correct identification of rotation and angle)
Total points check: 50.
Notes for grading and variations
- For questions with multiple-step coordinate work, award partial credit for correct intermediate steps.
- For Q9 a student could present an equivalent sequence (for example, translate first and then reflect across appropriately shifted line). Accept any correct sequence of rigid motions that produces the correct images; require explanation showing each image computed.
- If you prefer more multiple-choice items or diagram-based problems, tell me the grade level and I can produce a variant with figures instead of coordinate-only questions.
If you want, I can:
- Provide a printable PDF version.
- Replace any question with a diagram-based question (I can describe or give coordinates for drawing).
- Adjust difficulty for middle school, high-school geometry, or honors/IB level.
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