Geometry Quiz: Performing And Sequencing Rigid Transformations
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Performing And Sequencing Rigid TransformationsQuestion 1 of 11

Quadrilateral PQRSPQRS is transformed by a rotation of 90°90° counterclockwise about point T(3,1)T(3, 1) to create quadrilateral P′Q′R′S′P'Q'R'S'. If vertex PP has coordinates (7,3)(7, 3), what are the coordinates of P′P'?

(1,5)(1, 5)
(5,−1)(5, -1)
(−1,5)(-1, 5)
(1,−3)(1, -3)
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Geometry Quiz

Geometry Quiz: Performing And Sequencing Rigid Transformations

Practice Performing And Sequencing Rigid Transformations in Geometry 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 Performing And Sequencing Rigid Transformations, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

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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

Quadrilateral PQRSPQRS is transformed by a rotation of 90°90° counterclockwise about point T(3,1)T(3, 1) to create quadrilateral P′Q′R′S′P'Q'R'S'. If vertex PP has coordinates (7,3)(7, 3), what are the coordinates of P′P'?

  1. (1,5)(1, 5) (correct answer)
  2. (5,−1)(5, -1)
  3. (−1,5)(-1, 5)
  4. (1,−3)(1, -3)
Explanation: To rotate point P(7,3)P(7,3) by 90°90° counterclockwise about T(3,1)T(3,1): First, translate so TT is at origin: Ptemp=(7−3,3−1)=(4,2)P_{temp} = (7-3, 3-1) = (4,2). Then rotate 90°90° counterclockwise: (x,y)→(−y,x)(x,y) → (-y,x), so (4,2)→(−2,4)(4,2) → (-2,4). Finally, translate back: P′=(−2+3,4+1)=(1,5)P' = (-2+3, 4+1) = (1,5). Choice B uses clockwise rotation. Choice C forgets the final translation back. Choice D uses 180°180° rotation instead of 90°90°.

Question 2

Two congruent right triangles, △ABC\triangle ABC and △DEF\triangle DEF, are positioned in the coordinate plane such that they do not overlap. Triangle ABCABC has a right angle at CC, and triangle DEFDEF has a right angle at FF. If △ABC\triangle ABC can be mapped onto △DEF\triangle DEF using exactly two rigid transformations, and the first transformation is a reflection across the yy-axis, which statement about the second transformation must be true?

  1. The second transformation must be a translation that moves the reflected triangle horizontally only
  2. The second transformation must be a rotation about the origin to align corresponding vertices properly
  3. The second transformation could be any combination of rotation and translation that completes the mapping (correct answer)
  4. The second transformation must be another reflection across a line parallel to the xx-axis
Explanation: After reflecting △ABC\triangle ABC across the yy-axis, we get △A′B′C′\triangle A'B'C' which is congruent to the original but with opposite orientation (if we consider the order of vertices). To map this onto △DEF\triangle DEF, we need one more rigid transformation. This could be: (1) a translation if △A′B′C′\triangle A'B'C' and △DEF\triangle DEF have the same orientation and size, (2) a rotation if they have different orientations, or (3) a combination rotation and translation (which can be achieved as a single rotation about an appropriate point). The key insight is that any rigid transformation preserves distance and angles, so any single transformation that aligns the triangles will work. Choice A is too restrictive. Choice B assumes rotation about origin is required. Choice D would create a specific orientation that may not match △DEF\triangle DEF.

Question 3

On the coordinate plane, polygon RSTURSTU is mapped onto R′S′T′U′R'S'T'U'. Order matters. Which sequence correctly produces the image?

R(−1,1), S(2,1), T(2,2), U(−1,2)R(-1,1),\ S(2,1),\ T(2,2),\ U(-1,2) and R′(5,4), S′(5,1), T′(4,1), U′(4,4)R'(5,4),\ S'(5,1),\ T'(4,1),\ U'(4,4).

  1. Rotate 90∘90^\circ counterclockwise about the origin, then translate right 4 units and up 3 units.
  2. Rotate 90∘90^\circ clockwise about the origin, then translate right 4 units and up 3 units. (correct answer)
  3. Translate right 4 units and up 3 units, then rotate 90∘90^\circ clockwise about the origin.
  4. Reflect across the xx-axis, then translate right 4 units and up 3 units.
Explanation: This is a sequencing rigid transformations problem mapping polygon RSTU onto R'S'T'U'. The transformations needed are a rotation and a translation. The order matters - we rotate first to reorient, then translate to position. Rotating R(-1,1) 90° clockwise about the origin gives (1,1), then translating right 4 and up 3 gives (5,4) which matches R'. Similarly, S(2,1) rotates to (1,-2), then translates to (5,1) matching S'. The correct answer B works perfectly. Option C fails because translating before rotating would result in incorrect final positions due to the rotation affecting the translated coordinates.

Question 4

Square PQRSPQRS is rotated 45°45° counterclockwise about its center to create square P′Q′R′S′P'Q'R'S'. Then square P′Q′R′S′P'Q'R'S' is reflected across a line passing through the center of the original square to create square P′′Q′′R′′S′′P''Q''R''S''. If the final image P′′Q′′R′′S′′P''Q''R''S'' has the same orientation as the original square PQRSPQRS, what was the direction of the reflection line?

  1. The reflection line was parallel to one of the sides of the original square
  2. The reflection line was diagonal, making a 22.5°22.5° angle with the sides of the square (correct answer)
  3. The reflection line was perpendicular to one of the sides of the original square
  4. The reflection line was parallel to one of the diagonals of the original square
Explanation: After rotating 45°45° counterclockwise, the square's sides are at 45°45° angles to the original orientation. To return the square to its original orientation, the reflection must undo this 45°45° rotation. When reflecting across a line, the angle of rotation is twice the angle between the original orientation and the reflection line. To achieve a net rotation of −45°-45° (returning to original position), we need 2θ=45°2θ = 45°, so θ=22.5°θ = 22.5°. Therefore, the reflection line makes a 22.5°22.5° angle with the original sides of the square.

Question 5

In the coordinate plane, segment PQ‾\overline{PQ} is mapped onto segment P′Q′‾\overline{P'Q'}. Order matters. Which sequence correctly produces the image?

P(−4,2), Q(−1,5)P(-4,2),\ Q(-1,5) and P′(6,2), Q′(3,5)P'(6,2),\ Q'(3,5).

  1. Reflect across the yy-axis, then translate right 2 units. (correct answer)
  2. Translate right 2 units, then reflect across the yy-axis.
  3. Reflect across the xx-axis, then translate right 2 units.
  4. Reflect across the yy-axis.
Explanation: This is a sequencing rigid transformations problem where we need to map segment PQ onto segment P'Q'. The transformations needed are a reflection and a translation. The order of transformations matters - we must reflect first to change orientation, then translate to the final position. Reflecting P(-4,2) across the y-axis gives (4,2), then translating right 2 units gives (6,2) which matches P'. Similarly, Q(-1,5) reflects to (1,5), then translates to (3,5) matching Q'. The correct answer A works by fixing the orientation first through reflection. Option B fails because translating before reflecting would place the points in the wrong final positions.

Question 6

A triangle is shown in the coordinate plane as △ABC\triangle ABC and its image as △A′B′C′\triangle A'B'C'. Order matters. Which transformation must occur first to map △ABC\triangle ABC onto △A′B′C′\triangle A'B'C'?

A(−2,1), B(−5,2), C(−3,4)A(-2,1),\ B(-5,2),\ C(-3,4) and A′(2,−1), B′(5,−2), C′(3,−4)A'(2,-1),\ B'(5,-2),\ C'(3,-4).

  1. A translation 4 units to the right.
  2. A reflection across the yy-axis.
  3. A reflection across the origin (a 180∘180^\circ rotation about the origin). (correct answer)
  4. A rotation 90∘90^\circ clockwise about the origin.
Explanation: This is a sequencing rigid transformations problem where we need to identify the first transformation. Looking at the coordinates, A(-2,1) maps to A'(2,-1), B(-5,2) to B'(5,-2), and C(-3,4) to C'(3,-4). Each point (x,y) maps to (-x,-y), which is exactly a 180° rotation about the origin (or reflection through the origin). This single transformation completely maps the triangle - no additional transformations are needed. The correct answer C identifies this as a reflection across the origin. Option B (reflection across y-axis) would give (2,1) not (2,-1) for point A, so it fails.

Question 7

Refer to the diagram. Parallelogram ABCDABCD is transformed through a sequence of rigid transformations to create parallelogram A′B′C′D′A'B'C'D'. What is the minimum number of transformations needed to map ABCDABCD onto A′B′C′D′A'B'C'D'?

  1. One transformation: a single rotation about the point halfway between the parallelograms (correct answer)
  2. Two transformations: a reflection followed by a translation in the same direction
  3. Three transformations: a rotation, then reflection, then translation to achieve final position
  4. Two transformations: a translation followed by a rotation about vertex A′A'
Explanation: Looking at the relative positions and orientations of the parallelograms, A′B′C′D′A'B'C'D' appears to be rotated 180¬∞180¬∞ from ABCDABCD and displaced. A single 180¬∞180¬∞ rotation about the appropriate center point can map one parallelogram onto the other. Since both parallelograms have the same size and shape (congruent), and A′B′C′D′A'B'C'D' appears to be oriented opposite to ABCDABCD, a single rotation of 180¬∞180¬∞ about the midpoint between corresponding vertices is sufficient. Choice B would work but uses more transformations than necessary. Choice C is unnecessarily complex. Choice D could work but is not minimal since rotation alone can accomplish the mapping.

Question 8

On the coordinate plane, triangle RSTRST is mapped onto triangle R′S′T′R'S'T' using two rigid transformations. Order matters. Which description correctly orders the transformations?​

  1. Rotate 90∘90^\circ counterclockwise about the origin, then translate down 2 units. (correct answer)
  2. Translate down 2 units, then rotate 90∘90^\circ counterclockwise about the origin.
  3. Rotate 90∘90^\circ clockwise about the origin, then translate down 2 units.
  4. Translate down 2 units only.
Explanation: This is a sequencing rigid transformations problem where we need to map triangle RST onto triangle R'S'T'. The transformations needed are a rotation and a translation. The order matters because rotations about the origin affect both position and orientation. The correct sequence is to rotate 90° counterclockwise about the origin first, then translate down 2 units. This works because the rotation reorients the triangle and moves it to a new position around the origin, then the translation shifts it down to reach the final location. If we translated down first then rotated (choice B), the lowered triangle would follow a circular path during rotation and end up in the wrong position. The key principle is to perform rotations about the origin before translations to ensure the correct final placement.

Question 9

Triangle DEFDEF is mapped onto triangle D′E′F′D'E'F' on the coordinate plane by two rigid transformations. Order matters. Which description correctly orders the transformations?

  1. Rotate 90∘90^\circ clockwise about the origin, then translate up 3 units. (correct answer)
  2. Translate up 3 units, then rotate 90∘90^\circ clockwise about the origin.
  3. Rotate 90∘90^\circ counterclockwise about the origin, then translate up 3 units.
  4. Rotate 90∘90^\circ clockwise about the origin only.
Explanation: This is a sequencing rigid transformations problem where we need to map triangle DEF onto triangle D'E'F'. The transformations needed are a rotation and a translation. The order matters significantly because rotations about the origin change both orientation and position. The correct sequence is to rotate 90° clockwise about the origin first, then translate up 3 units. This works because the rotation reorients the triangle and moves it to a new position relative to the origin, then the translation shifts it up to the final location. If we translated up first then rotated (choice B), the triangle would end up in the wrong position because rotation about the origin would move the already-elevated triangle in a circular path. The key principle is to perform rotations about the origin before translations to achieve the desired final position.

Question 10

Triangle UVWUVW is mapped onto triangle U′V′W′U'V'W' on the coordinate plane by two rigid transformations. Order matters. Which sequence correctly produces the image?​

  1. Rotate 180∘180^\circ about the origin, then translate up 1 unit. (correct answer)
  2. Translate up 1 unit, then rotate 180∘180^\circ about the origin.
  3. Rotate 90∘90^\circ clockwise about the origin, then translate up 1 unit.
  4. Rotate 180∘180^\circ about the origin only.
Explanation: This is a sequencing rigid transformations problem where we need to map triangle UVW onto triangle U'V'W'. The transformations needed are a rotation and a translation. The order matters significantly because rotations about the origin change both orientation and position. The correct sequence is to rotate 180° about the origin first, then translate up 1 unit. This works because the 180° rotation flips the triangle to the opposite side of the origin with reversed orientation, then the translation moves it up slightly to the final position. If we translated up first then rotated (choice B), the elevated triangle would rotate to the wrong final position because the rotation would move it in a semicircle from its elevated starting point. The principle is to complete rotations about the origin before translations for predictable results.

Question 11

Polygon JKLMJKLM is mapped onto J′K′L′M′J'K'L'M' on the coordinate plane by two rigid transformations. Order matters. Which sequence correctly produces the image?​

  1. Reflect across the xx-axis, then rotate 180∘180^\circ about the origin.
  2. Rotate 180∘180^\circ about the origin, then reflect across the xx-axis. (correct answer)
  3. Rotate 90∘90^\circ clockwise about the origin, then reflect across the xx-axis.
  4. Rotate 180∘180^\circ about the origin only.
Explanation: This is a sequencing rigid transformations problem where we need to map polygon JKLM onto J'K'L'M'. The transformations needed are a rotation and a reflection. The order matters because these transformations don't commute - doing them in different orders produces different results. The correct sequence is to rotate 180° about the origin first, then reflect across the x-axis. This works because the 180° rotation flips the polygon to the opposite side of the origin, and then the reflection across the x-axis flips it vertically to achieve the final orientation and position. If we reflected first then rotated (choice A), we would get a different final position because the combination of these specific transformations in that order would place the polygon elsewhere. When combining rotations and reflections, the order significantly affects the final result.