Middle School Math Quiz: Understand Congruence Through Transformations

20 questions · exam conditions

0:00

Understand Congruence Through TransformationsQuestion 1 of 20

A polygon $JKLM$ has vertices $J(2,2)$ , $K(5,2)$ , $L(5,5)$ , and $M(2,5)$ . Another polygon $J'K'L'M'$ has vertices $J'(-2,-2)$ , $K'(-5,-2)$ , $L'(-5,-5)$ , and $M'(-2,-5)$ . Which transformation maps $JKLM$ to $J'K'L'M'$ ?

Reflect over the

y

-axis

Translate by

(-4,-4)

Rotate

180^\circ

about the origin

Reflect over the

x

-axis

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Understand Congruence Through Transformations

Practice Understand Congruence Through Transformations in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Understand Congruence Through Transformations, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Reflect over the $y$ -axis
Translate by $(-4,-4)$
Rotate $180^\circ$ about the origin (correct answer)
Reflect over the $x$ -axis

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For polygon JKLM with J(2,2), K(5,2), L(5,5), M(2,5) and J'K'L'M' with J'(-2,-2), K'(-5,-2), L'(-5,-5), M'(-2,-5), rotating 180° about origin maps (x,y) to (-x,-y): J to (-2,-2), K to (-5,-2), L to (-5,-5), M to (-2,-5), matching exactly. This 180° rotation is the correct rigid transformation that maps JKLM to J'K'L'M', confirming congruence. A common error is choosing a translation like (-4,-4), which would map J to (-2,-2) but K to (1,-2), not matching. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 2

Triangle $GHI$ has vertices $G(0,0)$ , $H(3,0)$ , and $I(0,2)$ . Triangle $JKL$ has vertices $J(0,0)$ , $K(6,0)$ , and $L(0,4)$ . Are the two triangles congruent using only rigid transformations (rotations, reflections, translations)?

No, because $JKL$ is a dilation of $GHI$ , not a rigid transformation. (correct answer)
Yes, because a rotation and translation can map $GHI$ onto $JKL$ .
No, because reflections are not allowed for congruence.
Yes, because a translation can map $GHI$ onto $JKL$ .

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. Here, GHI at (0,0),(3,0),(0,2) has sides 3,2,sqrt(13), while JKL at (0,0),(6,0),(0,4) has sides 6,4,sqrt(52)=2sqrt(13), showing a scale factor of 2, so only dilation maps them, not rigid transformations. Thus, they are not congruent, and the correct choice is C, as dilation is not rigid. A common error is claiming congruence via rotation and translation (choice B) without checking sizes differ. To find the sequence: (1) compare figures (sizes differ: bases 3 vs 6), (2) identify scaling, not just shift or turn, (3) no rigid sequence possible, (4) verification shows no exact match without scaling, (5) cannot simplify. Congruence means same size and shape via rigid transformations only, no scaling; common errors include ignoring size checks or confusing similarity with congruence.

Question 3

Polygon $JKLM$ has vertices $J(1,2)$ , $K(3,2)$ , $L(3,5)$ , and $M(1,5)$ . A student claims that rotating $JKLM$ $180^\circ$ about the origin maps it onto $J'K'L'M'$ with vertices $J'(-1,-2)$ , $K'(-3,-2)$ , $L'(-3,-5)$ , and $M'(-1,-5)$ . Is the student correct?

Yes, because a $180^\circ$ rotation about the origin sends $(x,y)$ to $(y,-x)$ .
No, because a $180^\circ$ rotation changes the size of the figure.
No, because a $180^\circ$ rotation about the origin sends $(x,y)$ to $(x,-y)$ .
Yes, because a $180^\circ$ rotation about the origin sends $(x,y)$ to $(-x,-y)$ . (correct answer)

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. The student's claim involves a 180° rotation about the origin, which sends (x,y) to (-x,-y): applying to J(1,2) gives (-1,-2), K(3,2) to (-3,-2), L(3,5) to (-3,-5), M(1,5) to (-1,-5), exactly matching the given points step-by-step. Therefore, the student is correct, as in choice A, since this rigid transformation maps the polygons congruently. A common error is thinking rotations change size or using incorrect mapping like (x,-y), which wouldn't align. To find the sequence: (1) compare sizes (match), (2) identify 180° turn, (3) build as rotation, (4) verify all points, (5) simplest; congruence excludes scaling, focuses on rigid motions.

Question 4

Triangle $ABC$ has vertices $A(1,0)$ , $B(3,0)$ , and $C(2,2)$ . Triangle $A'B'C'$ has vertices $A'(-1,4)$ , $B'(-3,4)$ , and $C'(-2,6)$ . Which sequence of rigid transformations maps triangle $ABC$ onto triangle $A'B'C'$ ?

Translate by $(-2,4)$ , then reflect over the $y$ -axis
Reflect over the $x$ -axis, then translate by $(0,4)$ (correct answer)
Rotate $180^\circ$ about the origin, then translate by $(-2,4)$
Reflect over the $y$ -axis, then translate by $(0,4)$

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. Here, first reflecting ABC at (1,0),(3,0),(2,2) over y-axis gives (-1,0),(-3,0),(-2,2), then translating by (0,4) yields (-1,4),(-3,4),(-2,6), matching A'B'C' step-by-step. Thus, choice B is the correct sequence, proving congruence. A common error is reversing order like choice A, which translates first then reflects, resulting in different positions like ( -3,4) not matching. To find the sequence: (1) compare figures (same size: base 2, etc.), (2) identify flip and shift, (3) build as reflection then translation, (4) verify sequential application matches all, (5) this order is necessary. Congruence means same size and shape via rigid transformations only, no scaling; common errors include wrong sequence order or omitting the reflection.

Question 5

Polygon $WXYZ$ has vertices $W(0,1)$ , $X(2,1)$ , $Y(2,4)$ , and $Z(0,4)$ . Polygon $W'X'Y'Z'$ has vertices $W'(3,2)$ , $X'(5,2)$ , $Y'(5,5)$ , and $Z'(3,5)$ . Are the two polygons congruent? If yes, which rigid transformation maps $WXYZ$ to $W'X'Y'Z'$ ?

Not congruent, because one figure is rotated.
Congruent; reflect over the $x$ -axis, then translate by $(3,1)$ .
Not congruent, because the side lengths change.
Congruent; translate by $(3,1)$ . (correct answer)

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via such transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other; for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation preserving size and shape, the triangles are congruent; if they had different sizes like sides 3-4-5 vs 6-8-10, they would not be congruent as that would require a dilation, which changes size. For polygons WXYZ at (0,1),(2,1),(2,4),(0,4) and W'X'Y'Z' at (3,2),(5,2),(5,5),(3,5), each point shifts by (3,1), mapping W to W', X to X', etc., exactly. Therefore, they are congruent via translation by (3,1), choice B. A common error is claiming not congruent due to position difference (choice A), ignoring translations preserve congruence. To find the sequence: (1) compare figures (same size: width 2, height 3), (2) identify just a shift, no flip or rotation, (3) build as translation (3,1), (4) verify applies to all vertices matching, (5) simplest single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include thinking rotations change congruence or not checking measurements.

Question 6

Triangle $ABC$ has vertices $A(1,2)$ , $B(4,2)$ , and $C(2,5)$ . Triangle $DEF$ has vertices $D(1,7)$ , $E(4,7)$ , and $F(2,10)$ . Which rigid transformation maps triangle $ABC$ onto triangle $DEF$ ?

Rotate $90^\circ$ counterclockwise about the origin
Translate by $(5,0)$
Reflect over the $x$ -axis, then translate by $(0,5)$
Translate by $(0,5)$ (correct answer)

Explanation: This question asks for the single rigid transformation that maps triangle

ABC

onto triangle

DEF

. Comparing vertices:

A(1,2)

D(1,7)

B(4,2)

E(4,7)

, and

C(2,5)

F(2,10)

each shift by the same amount, 0 units horizontally and 5 units vertically. Since every point moves by the same vector, this is a translation by

(0,5)

. It cannot be a rotation, since the triangle's orientation relative to the axes does not change, only its position. It cannot be a reflection either, because a reflection would flip the triangle, which does not happen here. Translate by

(5,0)

is incorrect because the shift is vertical, not horizontal.

Question 7

Triangle $ABC$ has vertices $A(0,0)$ , $B(4,0)$ , and $C(0,3)$ . Triangle $DEF$ has vertices $D(0,0)$ , $E(8,0)$ , and $F(0,6)$ . Are the two triangles congruent using only rigid transformations (translations, rotations, reflections)?

Yes, because a translation maps $ABC$ to $DEF$
Yes, because a reflection maps $ABC$ to $DEF$
No, because $DEF$ is a dilation of $ABC$ (different size) (correct answer)
No, because rotations are not allowed for congruence

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For triangle ABC with A(0,0), B(4,0), C(0,3) and DEF with D(0,0), E(8,0), F(0,6), the side lengths of ABC are AB=4, AC=3, BC=5, while DEF has DE=8, DF=6, EF=10, which is twice as large, so a dilation by factor 2 is needed, not just rigid transformations. Thus, the triangles are not congruent, as they have different sizes. A common error is claiming congruence via rotation when sizes differ, like option D which incorrectly states rotations are not allowed—they are, but size mismatch prevents congruence. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 8

Quadrilateral $WXYZ$ has vertices $W(1,1)$ , $X(4,1)$ , $Y(4,3)$ , and $Z(1,3)$ . Quadrilateral $W'X'Y'Z'$ has vertices $W'(-1,1)$ , $X'(-4,1)$ , $Y'(-4,3)$ , and $Z'(-1,3)$ . Which transformation maps $WXYZ$ to $W'X'Y'Z'$ ?

Reflect over the $x$ -axis
Reflect over the $y$ -axis (correct answer)
Rotate $180^\circ$ about the origin
Translate left 2 units

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For quadrilateral WXYZ with W(1,1), X(4,1), Y(4,3), Z(1,3) and W'X'Y'Z' with W'(-1,1), X'(-4,1), Y'(-4,3), Z'(-1,3), reflecting over the y-axis changes x to -x: W to (-1,1), X to (-4,1), Y to (-4,3), Z to (-1,3), matching exactly. This reflection is the correct rigid transformation that maps WXYZ to W'X'Y'Z', confirming congruence. A common error is selecting a rotation like 180° about the origin, which would map to different points such as W to (-1,-1), not matching. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 9

Pentagon $ABCDE$ has vertices $A(1,0)$ , $B(3,0)$ , $C(4,2)$ , $D(2,4)$ , and $E(0,2)$ . Pentagon $A'B'C'D'E'$ has vertices $A'(0,1)$ , $B'(0,3)$ , $C'(-2,4)$ , $D'(-4,2)$ , and $E'(-2,0)$ . Which rigid transformation maps $ABCDE$ to $A'B'C'D'E'$ ?

Rotate $180^\circ$ about the origin
Rotate $90^\circ$ clockwise about the origin
Rotate $90^\circ$ counterclockwise about the origin (correct answer)
Reflect over the line $y=x$

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For pentagon ABCDE with A(1,0), B(3,0), C(4,2), D(2,4), E(0,2) and A'B'C'D'E' with A'(0,1), B'(0,3), C'(-2,4), D'(-4,2), E'(-2,0), rotating 90° counterclockwise about origin: (x,y) to (-y,x), A(1,0)→(0,1), B(3,0)→(0,3), C(4,2)→(-2,4), D(2,4)→(-4,2), E(0,2)→(-2,0), matching exactly. This 90° counterclockwise rotation is the correct rigid transformation, confirming congruence. A common error is choosing 90° clockwise, which maps A(1,0) to (0,-1), not matching. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 10

Triangle $ABC$ has vertices $A(-2,1)$ , $B(1,1)$ , and $C(-1,4)$ . Triangle $DEF$ has vertices $D(3,1)$ , $E(6,1)$ , and $F(4,4)$ . Which rigid transformation maps triangle $ABC$ onto triangle $DEF$ ?

Translate up 5 units
Rotate $90^\circ$ clockwise
Translate right 5 units (correct answer)
Reflect over the $y$ -axis

Explanation: Comparing the two triangles, each vertex of DEF is exactly 5 more in the x-coordinate than the matching vertex of ABC, with the y-coordinate unchanged:

A(-2,1)\to D(3,1)

B(1,1)\to E(6,1)

C(-1,4)\to F(4,4)

. That's a translation right 5 units, so C is correct. Choice A is wrong because translating up would change the y-coordinates, not the x-coordinates, which is the opposite of what actually happened. Choice B is wrong because rotating

90°

clockwise about the origin would send

A(-2,1)

(1,2)

, which doesn't match

D(3,1)

. Choice D is wrong because reflecting over the y-axis would send

A(-2,1)

(2,1)

, which doesn't match

D(3,1)

either.

Question 11

Yes, because a translation maps $ABC$ to $DEF$
No, because $DEF$ is a dilation of $ABC$ (different size) (correct answer)
No, because rotations are not allowed for congruence
Yes, because a reflection maps $ABC$ to $DEF$

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For triangle ABC with A(0,0), B(4,0), C(0,3) and DEF with D(0,0), E(8,0), F(0,6), the side lengths of ABC are AB=4, AC=3, BC=5, while DEF has DE=8, DF=6, EF=10, which is twice as large, so a dilation by factor 2 is needed, not just rigid transformations. Thus, the triangles are not congruent, as they have different sizes. A common error is claiming congruence via rotation when sizes differ, like option D which incorrectly states rotations are not allowed—they are, but size mismatch prevents congruence. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 12

Rotate $180^\circ$ about the origin
Reflect over the $y$ -axis (correct answer)
Reflect over the $x$ -axis
Translate left 2 units

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For quadrilateral WXYZ with W(1,1), X(4,1), Y(4,3), Z(1,3) and W'X'Y'Z' with W'(-1,1), X'(-4,1), Y'(-4,3), Z'(-1,3), reflecting over the y-axis changes x to -x: W to (-1,1), X to (-4,1), Y to (-4,3), Z to (-1,3), matching exactly. This reflection is the correct rigid transformation that maps WXYZ to W'X'Y'Z', confirming congruence. A common error is selecting a rotation like 180° about the origin, which would map to different points such as W to (-1,-1), not matching. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 13

Rotate $90^\circ$ counterclockwise about the origin (correct answer)
Rotate $90^\circ$ clockwise about the origin
Rotate $180^\circ$ about the origin
Reflect over the line $y=x$

Explanation: This question tests understanding of congruent figures obtainable from each other by a sequence of rigid transformations (rotations, reflections, translations)—same size and shape means congruence via transformations. Two figures are congruent if a rigid transformation sequence maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—translation is a rigid transformation preserving size/shape, so the triangles are congruent; if different sizes (sides 3-4-5 vs 6-8-10), not congruent—would need dilation (scaling 2×) which isn't a rigid transformation (changes size); sequence description: identify transformations needed (flip? turn? shift?), order them (reflect first then translate, or rotate then reflect), verify maps all vertices correctly. For pentagon ABCDE with A(1,0), B(3,0), C(4,2), D(2,4), E(0,2) and A'B'C'D'E' with A'(0,1), B'(0,3), C'(-2,4), D'(-4,2), E'(-2,0), rotating 90° counterclockwise about origin: (x,y) to (-y,x), A(1,0)→(0,1), B(3,0)→(0,3), C(4,2)→(-2,4), D(2,4)→(-4,2), E(0,2)→(-2,0), matching exactly. This 90° counterclockwise rotation is the correct rigid transformation, confirming congruence. A common error is choosing 90° clockwise, which maps A(1,0) to (0,-1), not matching. To find the sequence: (1) compare figures (same size? check side lengths, angles), (2) identify orientation difference (flipped? rotated? just shifted?), (3) build sequence (if flipped: reflection needed, if rotated: rotation needed, if different position: translation), (4) verify (apply transformations to figure 1, should get figure 2 exactly—all vertices match), (5) simplify if possible (fewest transformations needed). Congruence means same size and shape, obtainable by rigid transformations only (rotation/reflection/translation), no scaling/stretching/skewing; common errors include including dilation (that's similarity), wrong order giving wrong final position, incomplete sequence (missing a needed transformation), or claiming congruence when sizes differ (not checking all measurements).

Question 14

Triangle $ABC$ has vertices $A(1,2)$ , $B(4,2)$ , and $C(2,5)$ . Triangle $A'B'C'$ has vertices $A'(3,4)$ , $B'(6,4)$ , and $C'(4,7)$ . Which statement is true?

The triangles are congruent because a reflection over the $x$ -axis maps $ABC$ to $A'B'C'$ .
The triangles are not congruent because translations change side lengths.
The triangles are not congruent because the second triangle is in Quadrant I.
The triangles are congruent because a translation by $(2,2)$ maps $ABC$ to $A'B'C'$ . (correct answer)

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. Adding (2,2) to each point maps ABC to A'B'C': (1,2) to (3,4), (4,2) to (6,4), (2,5) to (4,7), a translation confirming congruence. Therefore, choice A is correct, as the triangles are congruent via this rigid transformation. A common error is thinking translations change lengths or claiming non-congruence due to position. To find the sequence: (1) compare (same size), (2) identify shift, (3) build translation (2,2), (4) verify matches, (5) simplest; rigid motions define congruence, not location or orientation alone.

Question 15

Two triangles are shown by their coordinates: $\triangle ABC$ has $A(0,0)$ , $B(4,0)$ , $C(1,3)$ and $\triangle DEF$ has $D(0,0)$ , $E(8,0)$ , $F(2,6)$ . Are the triangles congruent using only rotations, reflections, and translations?

Yes, because triangle $DEF$ is a translation of triangle $ABC$ .
No, because mapping $ABC$ to $DEF$ would require a dilation (the size changes). (correct answer)
No, because reflections and rotations cannot change the orientation of a triangle.
Yes, because triangle $DEF$ is a reflection of triangle $ABC$ over the $x$ -axis.

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. Here, ABC has base 4 units and height 3, while DEF has base 8 and height 6, showing a scale factor of 2, so no rigid transformation maps them; distances like AB=4 vs DE=8 confirm different sizes. Thus, they are not congruent, as in choice C, since dilation is required. A common error is claiming congruence without checking sizes or thinking reflections change orientation in a way that affects congruence. To find the sequence: (1) compare (sizes differ), (2) identify scaling, (3) cannot build rigid sequence, (4) verification fails, (5) no simplification; congruence requires exact match via rigid motions only.

Question 16

In the coordinate plane, parallelogram KLMN is congruent to parallelogram PQRS. A student claims that a 180° rotation about the origin maps KLMN to PQRS. How can you verify whether this claim is correct?

Check if the distance from the origin to each vertex of KLMN equals the distance from the origin to the corresponding vertex of PQRS
Check if each vertex of PQRS is located at the point (-x, -y) where (x, y) is the coordinate of the corresponding vertex in KLMN (correct answer)
Check if the slopes of corresponding sides in both parallelograms are equal, since rotation preserves slope relationships
Check if the midpoint of the segment connecting each vertex of KLMN to its corresponding vertex in PQRS is the origin

Explanation: A 180° rotation about the origin maps each point (x, y) to (-x, -y). To verify the student's claim, we need to check if each vertex of PQRS is at the location (-x, -y) where (x, y) is the corresponding vertex in KLMN. Choice A only verifies that vertices are equidistant from the origin, but doesn't check the specific 180° rotation relationship. Choice C is incorrect because 180° rotation changes slopes (unless they're horizontal/vertical). Choice D checks for a different type of rotation center.

Question 17

Consider the two congruent octagons shown in the figure. Octagon ABCDEFGH can be mapped to octagon PQRSTUVW through a sequence of transformations. If the mapping uses the minimum possible number of transformations, and both octagons have the same orientation, what is the maximum number of transformations required?

One transformation (correct answer)
Two transformations
Three transformations
Four transformations

Explanation: When two congruent figures have the same orientation, no reflection is needed, since a reflection would flip the orientation. Any combination of rotations and translations that maps one figure onto the other can always be simplified into a single rotation or a single translation, so one transformation is always enough. Choice B is wrong because a rotation and a translation together can always be combined into one equivalent single transformation; two are never required. Choice C is wrong because reflections aren't needed here, and no extra transformations are required beyond that single rotation or translation. Choice D is wrong because the number of sides on the octagons has no effect on how many transformations are needed.

Question 18

Rotate $180^\circ$ about the origin (correct answer)
Translate by $(-4,-4)$
Reflect over the $x$ -axis
Reflect over the $y$ -axis

Question 19

Pentagon $ABCDE$ has vertices $A(1,1)$ , $B(3,1)$ , $C(4,2)$ , $D(2,4)$ , and $E(0,2)$ . Pentagon $A'B'C'D'E'$ has vertices $A'(1,-1)$ , $B'(3,-1)$ , $C'(4,-2)$ , $D'(2,-4)$ , and $E'(0,-2)$ . Which transformation maps $ABCDE$ onto $A'B'C'D'E'$ ?

Rotate $90^\circ$ clockwise about the origin
Reflect over the $y$ -axis
Reflect over the $x$ -axis (correct answer)
Translate by $(0,-2)$

Explanation: This question tests understanding of congruent figures that can be obtained from each other by a sequence of rigid transformations (rotations, reflections, translations)—having the same size and shape means they are congruent via these transformations. Two figures are congruent if there is a sequence of rigid transformations that maps one to the other: for example, triangle ABC with vertices (1,1),(4,1),(2,3) maps to triangle DEF at (1,5),(4,5),(2,7) via translation by (0,4)—since translation is a rigid transformation that preserves size and shape, the triangles are congruent. If the figures had different sizes, such as sides 3-4-5 versus 6-8-10, they would not be congruent because that would require a dilation (scaling by 2), which isn't a rigid transformation as it changes the size. For the pentagon, each point's y-coordinate is negated while x remains: A(1,1) to (1,-1), B(3,1) to (3,-1), up to E(0,2) to (0,-2), a reflection over the x-axis mapping exactly step-by-step. Therefore, choice A is correct, confirming congruence via this rigid transformation. A common error is using wrong transformation like rotation that doesn't fit or including dilation unnecessarily. To find the sequence: (1) compare (same size), (2) identify flip over x-axis, (3) build reflection, (4) verify all vertices, (5) simplest; congruence excludes non-rigid changes like scaling.

Question 20

Based on the coordinate plane shown, quadrilateral PQRS is congruent to quadrilateral WXYZ. If a single transformation maps PQRS to WXYZ, which statement about the transformation is correct?

The transformation is a reflection, and the line of reflection passes through the midpoint of each corresponding vertex pair
The transformation is a rotation, and the center of rotation is equidistant from all corresponding vertex pairs (correct answer)
The transformation is a translation, and the translation vector connects each vertex to its corresponding vertex
The transformation preserves orientation, so it cannot be a reflection across any line in the coordinate plane

Explanation: For a single transformation to map one figure to a congruent figure, if it's a rotation, the center of rotation must be equidistant from corresponding points. Choice A is incorrect because the line of reflection would be the perpendicular bisector of segments connecting corresponding vertices, not necessarily passing through midpoints. Choice C describes the translation vector but doesn't specify that this works only for translations. Choice D incorrectly assumes the orientation is preserved without seeing the actual figures.