Asked by .
Transformations and Congruence Unit Test
All Answers 1
Answered by
GPT-5 mini
AI
Transformations and Congruence — Unit Test
Directions: Answer all questions. Show all work for coordinate and proof problems. You may use a calculator for arithmetic only where needed. Label all transformations and indicate order when composing.
Section I — Multiple Choice (1 point each)
Circle the best answer.
1. Which of the following is NOT a rigid motion?
A. translation
B. rotation
C. reflection
D. dilation with scale factor 2
Answer: D
2. Which single transformation always preserves orientation (i.e., preserves the clockwise/counterclockwise order of points)?
A. reflection
B. rotation
C. dilation
D. none of the above
Answer: B
3. The image of point (−4, 7) after reflection across the x-axis is:
A. (4, −7)
B. (−4, −7)
C. (4, 7)
D. (−7, 4)
Answer: B
4. Under a 90° counterclockwise rotation about the origin, the point (2, −3) maps to:
A. (3, 2)
B. (−3, −2)
C. (−2, 3)
D. (3, −2)
Answer: A
5. Under dilation with center (0,0) and scale factor 1/3, the point (9, −6) maps to:
A. (3, −2)
B. (27, −18)
C. (−3, 2)
D. (9, −2)
Answer: A
6. If rigid motion maps A → A', B → B', C → C', then the side BC corresponds to:
A. A'B'
B. B'C'
C. C'A'
D. A'C'
Answer: B
7. Which transformation maps triangle with vertices (0,0), (2,0), (0,2) to (1,1), (3,1), (1,3)?
A. rotation 90° about origin
B. translation by vector (1,1)
C. dilation about origin by k = 2
D. reflection across line y = x
Answer: B
8. Do the following in order: reflect across the x-axis, then rotate 90° CCW about the origin. The image of (1,2) is:
A. (−2, −1)
B. (2,1)
C. (−1,2)
D. (1, −2)
Answer: B
9. A dilation with center (1,1) and scale factor 2 sends (3,5) to:
A. (6,10)
B. (5,9)
C. (7,11)
D. (4,8)
Answer: B
10. Which of the following does NOT prove two triangles are congruent?
A. SSS
B. SAS
C. ASA
D. AAA
Answer: D
Section II — Short Answer / Coordinate Work (2–3 points each)
11. Rotate triangle with vertices A(1,2), B(4,6), C(5,2) by 180° about the origin. Give coordinates of A', B', C'.
Answer: 180° about origin: (x,y) → (−x, −y)
A' = (−1, −2), B' = (−4, −6), C' = (−5, −2).
12. A dilation about the origin sends (2,3) to (6,9). What is the scale factor?
Answer: k = 6/2 = 9/3 = 3.
13. Find the translation vector that maps triangle with vertices (0,0), (3,0), (0,4) to (5,1), (8,1), (5,5).
Answer: Translation vector v = (5,1) (add (5,1) to each original vertex).
14. Give the equation of the image of the line y = 2x + 1 after reflecting across the y-axis.
Answer: Reflection across y-axis: x → −x, so y = 2(−x) + 1 = −2x + 1.
15. True or False: A dilation with scale factor k = −2 can be described as a dilation by 2 combined with a 180° rotation about the center. Briefly justify.
Answer: True. Multiplying all position vectors by −2 equals multiplying by 2 (dilation) and by −1, which is equivalent to a 180° rotation about the center.
Section III — Problem Solving / Composition (3–4 points each)
16. Triangle ABC has A(1,1), B(4,1), C(1,5). Rotate the triangle 90° CCW about the origin. Give coordinates of A', B', C' and briefly explain why ΔABC ≅ ΔA'B'C'.
Work/Answer:
Rotation 90° CCW: (x,y) → (−y, x)
A' = (−1, 1)
B' = (−1, 4)
C' = (−5, 1)
Reason: Rotation is a rigid motion (preserves all distances and angles), so the image triangle is congruent to the original.
17. Reflect Δ with vertices A(−2,3), B(1,3), C(−2, −1) across the line y = x. Give coordinates A', B', C'.
Answer: Reflection across y = x swaps coordinates: (x,y) → (y,x)
A' = (3, −2)
B' = (3, 1)
C' = (−1, −2)
18. Consider the transformation T which is: first dilate about the origin by k = 3, then reflect across the x-axis. Find the inverse T^−1 and apply it to the point (6, −9). Show steps.
Work/Answer:
T = (reflect across x-axis) ◦ (dilation k=3 about origin).
Inverse reverses order and uses inverse of each:
- inverse of reflection across x-axis is itself (reflect across x-axis).
- inverse of dilation k = 3 is dilation with k = 1/3 about same center (origin).
So T^−1 = (dilation k = 1/3 about origin) ◦ (reflect across x-axis).
Apply to (6, −9):
Step 1 reflect across x-axis: (6, −9) → (6, 9).
Step 2 dilate by 1/3 about origin: (6, 9) → (2, 3).
So T^−1(6, −9) = (2, 3).
19. Given triangle with vertices P(2,2), Q(6,2), R(2,8), perform a dilation of scale factor 1/2 centered at (2,2). Find P', Q', R'.
Work/Answer:
Center (2,2) fixed: P is center so P' = (2,2).
For Q: vector from center to Q = (4,0); half is (2,0); Q' = center + (2,0) = (4,2).
For R: vector (0,6); half (0,3); R' = (2,5).
So P' = (2,2), Q' = (4,2), R' = (2,5).
Section IV — Proof / Justification (6 points)
20. Given triangles ΔABC and ΔA'B'C' with AB = A'B', AC = A'C', and ∠BAC = ∠B'A'C', prove ΔABC ≅ ΔA'B'C'. Then state which corresponding sides are equal by CPCTC.
Proof/Answer:
By the given, two sides and the included angle of ΔABC are equal to two sides and the included angle of ΔA'B'C' (AB = A'B', AC = A'C', and ∠BAC = ∠B'A'C'). Therefore, by SAS (Side-Angle-Side) congruence, ΔABC ≅ ΔA'B'C'. By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the remaining corresponding sides and angles are equal; in particular, BC = B'C' (and ∠ABC = ∠A'B'C', ∠ACB = ∠A'C'B').
--- End of Test ---
Answer Key (concise)
Section I: 1:D 2:B 3:B 4:A 5:A 6:B 7:B 8:B 9:B 10:D
Section II: 11: A'(−1,−2), B'(−4,−6), C'(−5,−2)
12: k = 3
13: (5,1)
14: y = −2x + 1
15: True (−2 = 2·(−1), factor −1 = 180° rotation)
Section III: 16: A'(−1,1), B'(−1,4), C'(−5,1); rotation preserves distances ⇒ congruent
17: A'(3,−2), B'(3,1), C'(−1,−2)
18: T^−1 = (dilation by 1/3 about origin) ◦ (reflection across x-axis); T^−1(6,−9) = (2,3)
19: P'(2,2), Q'(4,2), R'(2,5)
Section IV: 20: SAS ⇒ congruent; by CPCTC, BC = B'C' (and other corresponding parts equal)
If you want this formatted as a printable PDF (with room for student answers), or a version tailored to a specific grade level or standard (e.g., Common Core / state standard references), tell me which format or standards and I’ll prepare it.
Directions: Answer all questions. Show all work for coordinate and proof problems. You may use a calculator for arithmetic only where needed. Label all transformations and indicate order when composing.
Section I — Multiple Choice (1 point each)
Circle the best answer.
1. Which of the following is NOT a rigid motion?
A. translation
B. rotation
C. reflection
D. dilation with scale factor 2
Answer: D
2. Which single transformation always preserves orientation (i.e., preserves the clockwise/counterclockwise order of points)?
A. reflection
B. rotation
C. dilation
D. none of the above
Answer: B
3. The image of point (−4, 7) after reflection across the x-axis is:
A. (4, −7)
B. (−4, −7)
C. (4, 7)
D. (−7, 4)
Answer: B
4. Under a 90° counterclockwise rotation about the origin, the point (2, −3) maps to:
A. (3, 2)
B. (−3, −2)
C. (−2, 3)
D. (3, −2)
Answer: A
5. Under dilation with center (0,0) and scale factor 1/3, the point (9, −6) maps to:
A. (3, −2)
B. (27, −18)
C. (−3, 2)
D. (9, −2)
Answer: A
6. If rigid motion maps A → A', B → B', C → C', then the side BC corresponds to:
A. A'B'
B. B'C'
C. C'A'
D. A'C'
Answer: B
7. Which transformation maps triangle with vertices (0,0), (2,0), (0,2) to (1,1), (3,1), (1,3)?
A. rotation 90° about origin
B. translation by vector (1,1)
C. dilation about origin by k = 2
D. reflection across line y = x
Answer: B
8. Do the following in order: reflect across the x-axis, then rotate 90° CCW about the origin. The image of (1,2) is:
A. (−2, −1)
B. (2,1)
C. (−1,2)
D. (1, −2)
Answer: B
9. A dilation with center (1,1) and scale factor 2 sends (3,5) to:
A. (6,10)
B. (5,9)
C. (7,11)
D. (4,8)
Answer: B
10. Which of the following does NOT prove two triangles are congruent?
A. SSS
B. SAS
C. ASA
D. AAA
Answer: D
Section II — Short Answer / Coordinate Work (2–3 points each)
11. Rotate triangle with vertices A(1,2), B(4,6), C(5,2) by 180° about the origin. Give coordinates of A', B', C'.
Answer: 180° about origin: (x,y) → (−x, −y)
A' = (−1, −2), B' = (−4, −6), C' = (−5, −2).
12. A dilation about the origin sends (2,3) to (6,9). What is the scale factor?
Answer: k = 6/2 = 9/3 = 3.
13. Find the translation vector that maps triangle with vertices (0,0), (3,0), (0,4) to (5,1), (8,1), (5,5).
Answer: Translation vector v = (5,1) (add (5,1) to each original vertex).
14. Give the equation of the image of the line y = 2x + 1 after reflecting across the y-axis.
Answer: Reflection across y-axis: x → −x, so y = 2(−x) + 1 = −2x + 1.
15. True or False: A dilation with scale factor k = −2 can be described as a dilation by 2 combined with a 180° rotation about the center. Briefly justify.
Answer: True. Multiplying all position vectors by −2 equals multiplying by 2 (dilation) and by −1, which is equivalent to a 180° rotation about the center.
Section III — Problem Solving / Composition (3–4 points each)
16. Triangle ABC has A(1,1), B(4,1), C(1,5). Rotate the triangle 90° CCW about the origin. Give coordinates of A', B', C' and briefly explain why ΔABC ≅ ΔA'B'C'.
Work/Answer:
Rotation 90° CCW: (x,y) → (−y, x)
A' = (−1, 1)
B' = (−1, 4)
C' = (−5, 1)
Reason: Rotation is a rigid motion (preserves all distances and angles), so the image triangle is congruent to the original.
17. Reflect Δ with vertices A(−2,3), B(1,3), C(−2, −1) across the line y = x. Give coordinates A', B', C'.
Answer: Reflection across y = x swaps coordinates: (x,y) → (y,x)
A' = (3, −2)
B' = (3, 1)
C' = (−1, −2)
18. Consider the transformation T which is: first dilate about the origin by k = 3, then reflect across the x-axis. Find the inverse T^−1 and apply it to the point (6, −9). Show steps.
Work/Answer:
T = (reflect across x-axis) ◦ (dilation k=3 about origin).
Inverse reverses order and uses inverse of each:
- inverse of reflection across x-axis is itself (reflect across x-axis).
- inverse of dilation k = 3 is dilation with k = 1/3 about same center (origin).
So T^−1 = (dilation k = 1/3 about origin) ◦ (reflect across x-axis).
Apply to (6, −9):
Step 1 reflect across x-axis: (6, −9) → (6, 9).
Step 2 dilate by 1/3 about origin: (6, 9) → (2, 3).
So T^−1(6, −9) = (2, 3).
19. Given triangle with vertices P(2,2), Q(6,2), R(2,8), perform a dilation of scale factor 1/2 centered at (2,2). Find P', Q', R'.
Work/Answer:
Center (2,2) fixed: P is center so P' = (2,2).
For Q: vector from center to Q = (4,0); half is (2,0); Q' = center + (2,0) = (4,2).
For R: vector (0,6); half (0,3); R' = (2,5).
So P' = (2,2), Q' = (4,2), R' = (2,5).
Section IV — Proof / Justification (6 points)
20. Given triangles ΔABC and ΔA'B'C' with AB = A'B', AC = A'C', and ∠BAC = ∠B'A'C', prove ΔABC ≅ ΔA'B'C'. Then state which corresponding sides are equal by CPCTC.
Proof/Answer:
By the given, two sides and the included angle of ΔABC are equal to two sides and the included angle of ΔA'B'C' (AB = A'B', AC = A'C', and ∠BAC = ∠B'A'C'). Therefore, by SAS (Side-Angle-Side) congruence, ΔABC ≅ ΔA'B'C'. By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), the remaining corresponding sides and angles are equal; in particular, BC = B'C' (and ∠ABC = ∠A'B'C', ∠ACB = ∠A'C'B').
--- End of Test ---
Answer Key (concise)
Section I: 1:D 2:B 3:B 4:A 5:A 6:B 7:B 8:B 9:B 10:D
Section II: 11: A'(−1,−2), B'(−4,−6), C'(−5,−2)
12: k = 3
13: (5,1)
14: y = −2x + 1
15: True (−2 = 2·(−1), factor −1 = 180° rotation)
Section III: 16: A'(−1,1), B'(−1,4), C'(−5,1); rotation preserves distances ⇒ congruent
17: A'(3,−2), B'(3,1), C'(−1,−2)
18: T^−1 = (dilation by 1/3 about origin) ◦ (reflection across x-axis); T^−1(6,−9) = (2,3)
19: P'(2,2), Q'(4,2), R'(2,5)
Section IV: 20: SAS ⇒ congruent; by CPCTC, BC = B'C' (and other corresponding parts equal)
If you want this formatted as a printable PDF (with room for student answers), or a version tailored to a specific grade level or standard (e.g., Common Core / state standard references), tell me which format or standards and I’ll prepare it.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.