- 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?
Answer: One transformation
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.
- 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?
Answer: Translate right 5 units
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)→D(3,1), B(1,1)→E(6,1), C(−1,4)→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) to (1,2), which doesn't match D(3,1). Choice D is wrong because reflecting over the y-axis would send A(−2,1) to (2,1), which doesn't match D(3,1) either. - 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?
Answer: Translate by (0,5)
This question asks for the single rigid transformation that maps triangle ABC onto triangle DEF. Comparing vertices: A(1,2) to D(1,7), B(4,2) to E(4,7), and C(2,5) to 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. - A regular pentagon ABCDE is congruent to regular pentagon FGHIJ. If pentagon ABCDE is rotated 72° clockwise about its center, the result is identical to pentagon ABCDE. When pentagon FGHIJ is mapped to pentagon ABCDE, which statement about the transformation sequence is most accurate?
Answer: The transformation sequence must include at least one translation or reflection, since rotational symmetry alone cannot map between different pentagons
While both pentagons have the same rotational symmetry (72° rotations), this internal symmetry doesn't determine how to map one pentagon to another. The pentagons are in different locations and possibly different orientations, so the mapping requires transformations to change position and/or orientation. A rotation about the center of one pentagon cannot move it to the location of the other pentagon. Choice A incorrectly assumes rotational symmetry determines the mapping. Choice B incorrectly suggests any rotation works. Choice D ignores the need to change position.
- Polygon JKLM has vertices J(1,2), K(3,2), L(3,5), and M(1,5). A student claims that rotating JKLM 180∘ 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?
Answer: Yes, because a 180∘ rotation about the origin sends (x,y) to (−x,−y).
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. - 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′?
Answer: Reflect over the x-axis, then translate by (0,4)
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. - 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′?
Answer: Rotate 180∘ about the origin
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). - 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′?
Answer: Rotate 90∘ counterclockwise about the origin
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). - 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?
Answer: The transformation is a rotation, and the center of rotation is equidistant from all corresponding vertex pairs
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.
- Triangle ABC has vertices A(2,1), B(5,1), and C(3,4). Triangle A′B′C′ has vertices A′(2,−1), B′(5,−1), and C′(3,−4). Which transformation maps triangle ABC onto triangle A′B′C′?
Answer: Reflect over the x-axis
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 triangle ABC at A(2,1), B(5,1), C(3,4) and A'(2,-1), B'(5,-1), C'(3,-4), negating y-coordinates while keeping x the same is a reflection over the x-axis, mapping each vertex accordingly in one step. Thus, the correct choice is B, reflection over the x-axis, proving congruence with reversed orientation. A common error is selecting translation like (0,-2) (choice C), which shifts but doesn't flip, resulting in mismatch like (2,-1) but others not aligning fully. To find the sequence: (1) compare figures (same size: AB=3, A'B'=3), (2) identify flipped over horizontal axis, (3) build as reflection over x-axis, (4) verify application matches all vertices, (5) simplest as single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include claiming non-congruence for orientation differences or using rotation instead of reflection. - 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′?
Answer: Congruent; translate by (3,1).
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. - Quadrilateral PQRS has vertices P(1,1), Q(4,1), R(4,3), and S(1,3). Quadrilateral P′Q′R′S′ has vertices P′(−1,1), Q′(−4,1), R′(−4,3), and S′(−1,3). Which transformation maps PQRS onto P′Q′R′S′?
Answer: Reflect over the y-axis
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, quadrilateral PQRS at P(1,1), Q(4,1), R(4,3), S(1,3) maps to P'(-1,1), Q'(-4,1), R'(-4,3), S'(-1,3) by negating x-coordinates while keeping y the same, which is a reflection over the y-axis applied step-by-step to each vertex. Therefore, the correct transformation is reflection over the y-axis, choice C, confirming congruence. A common error is choosing rotation like 180° about origin (choice B), which would map (1,1) to (-1,-1) not matching, or ignoring the flip in orientation. To find the sequence: (1) compare figures (same size: sides like PQ=3, P'Q'=3), (2) identify flipped over vertical axis, (3) build as reflection over y-axis, (4) verify applying to all points matches exactly, (5) simplest single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include using dilation for size changes or incorrect order leading to mismatch. - 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′?
Answer: Rotate 90∘ counterclockwise about the origin
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). - Triangle ABC has vertices A(2,3), B(6,3), and C(4,6). Triangle DEF has vertices D(−2,−3), E(−6,−3), and F(−4,−6). Which transformation maps triangle ABC onto triangle DEF?
Answer: Rotate 180∘ about the origin
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 ABC at (2,3),(6,3),(4,6) to DEF at (-2,-3),(-6,-3),(-4,-6), applying 180° rotation about origin negates both coordinates: (2,3) to (-2,-3), etc., mapping exactly. Therefore, choice A is correct, confirming congruence via rotation. A common error is selecting translation like (-4,-6) (choice C), which would map (2,3) to (-2,-3) but (6,3) to (2,-3) not matching. To find the sequence: (1) compare figures (same size: base 4, etc.), (2) identify 180° turn, (3) build as rotation about origin, (4) verify all points match, (5) simplest single transformation. Congruence means same size and shape via rigid transformations only, no scaling; common errors include incomplete sequences or wrong center of rotation. - 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)?
Answer: No, because DEF is a dilation of ABC (different size)
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). - Triangle ABC has vertices A(1,1), B(5,1), and C(2,4). Triangle DEF has vertices D(1,6), E(5,6), and F(2,9). Which rigid transformation maps triangle ABC onto triangle DEF?
Answer: Translate by (0,5)
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. In this case, starting with A(1,1) to D(1,6), B(5,1) to E(5,6), and C(2,4) to F(2,9), each point shifts up by 5 units in the y-direction, so a translation by (0,5) maps ABC exactly onto DEF step-by-step: A becomes (1,6), B (5,6), C (2,9). Therefore, the correct choice is B, translate by (0,5), confirming congruence through this rigid motion. A common error is including dilation for congruence, which only creates similarity, not congruence, or choosing a wrong transformation like rotation that doesn't align the vertices. To find the sequence: (1) compare figures (same size, check side lengths like AB=4, DE=4), (2) identify just a vertical shift, (3) build as translation (0,5), (4) verify all vertices match, (5) simplest with one transformation; remember congruence requires only rigid motions, no scaling. - Triangle RST has vertices R(−2,1), S(−5,1), and T(−3,4). Triangle R′S′T′ has vertices R′(2,1), S′(5,1), and T′(3,4). Which transformation maps RST onto R′S′T′?
Answer: Reflect over the y-axis
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 triangles map by negating x-coordinates while keeping y: R(-2,1) to (2,1), S(-5,1) to (5,1), T(-3,4) to (3,4), which is a reflection over the y-axis. So, choice B is correct, showing congruence through this rigid motion. A common error is choosing rotation or translation that doesn't align points or claiming size change. To find the sequence: (1) compare (same size), (2) identify flip over y-axis, (3) build reflection, (4) verify matches, (5) simplest; rigid transformations preserve size, unlike dilations for similarity. - 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′?
Answer: Reflect over the y-axis
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). - 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)?
Answer: No, because DEF is a dilation of ABC (different size)
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). - 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′?
Answer: Reflect over the x-axis
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.
