Performing and Sequencing Rigid Transformations - Geometry

1. A triangle is shown in the coordinate plane as $\triangle ABC$ and its image as $\triangle A'B'C'$. Order matters. Which transformation must occur first to map $\triangle ABC$ onto $\triangle A'B'C'$? $A(-2,1),\ B(-5,2),\ C(-3,4)$ and $A'(2,-1),\ B'(5,-2),\ C'(3,-4)$.
   Answer: A reflection across the origin (a $180^\circ$ rotation about the origin). 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.
2. On the coordinate plane, triangle $RST$ is mapped onto triangle $R'S'T'$ using two rigid transformations. Order matters. Which description correctly orders the transformations?​
   Answer: Rotate $90^\circ$ counterclockwise about the origin, then translate down 2 units. 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.
3. Two congruent right triangles, $\triangle ABC$ and $\triangle DEF$, are positioned in the coordinate plane such that they do not overlap. Triangle $ABC$ has a right angle at $C$, and triangle $DEF$ has a right angle at $F$. If $\triangle ABC$ can be mapped onto $\triangle DEF$ using exactly two rigid transformations, and the first transformation is a reflection across the $y$-axis, which statement about the second transformation must be true?
   Answer: The second transformation could be any combination of rotation and translation that completes the mapping After reflecting $\triangle ABC$ across the $y$-axis, we get $\triangle A'B'C'$ which is congruent to the original but with opposite orientation (if we consider the order of vertices). To map this onto $\triangle DEF$, we need one more rigid transformation. This could be: (1) a translation if $\triangle A'B'C'$ and $\triangle DEF$ 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 $\triangle DEF$.
4. On the coordinate plane, polygon $RSTU$ is mapped onto $R'S'T'U'$. Order matters. Which sequence correctly produces the image? $R(-1,1),\ S(2,1),\ T(2,2),\ U(-1,2)$ and $R'(5,4),\ S'(5,1),\ T'(4,1),\ U'(4,4)$.
   Answer: Rotate $90^\circ$ clockwise about the origin, then translate right 4 units and up 3 units. 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.
5. In the coordinate plane, segment $\overline{PQ}$ is mapped onto segment $\overline{P'Q'}$. Order matters. Which sequence correctly produces the image? $P(-4,2),\ Q(-1,5)$ and $P'(6,2),\ Q'(3,5)$.
   Answer: Reflect across the $y$-axis, then translate right 2 units. 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.
6. Quadrilateral $PQRS$ is transformed by a rotation of $90°$ counterclockwise about point $T(3, 1)$ to create quadrilateral $P'Q'R'S'$. If vertex $P$ has coordinates $(7, 3)$, what are the coordinates of $P'$?
   Answer: $(1, 5)$ To rotate point $P(7,3)$ by $90°$ counterclockwise about $T(3,1)$: First, translate so $T$ is at origin: $P_{temp} = (7-3, 3-1) = (4,2)$. Then rotate $90°$ counterclockwise: $(x,y) → (-y,x)$, so $(4,2) → (-2,4)$. Finally, translate back: $P' = (-2+3, 4+1) = (1,5)$. Choice B uses clockwise rotation. Choice C forgets the final translation back. Choice D uses $180°$ rotation instead of $90°$.
7. Refer to the diagram. Parallelogram $ABCD$ is transformed through a sequence of rigid transformations to create parallelogram $A'B'C'D'$. What is the minimum number of transformations needed to map $ABCD$ onto $A'B'C'D'$?
   Answer: One transformation: a single rotation about the point halfway between the parallelograms Looking at the relative positions and orientations of the parallelograms, $A'B'C'D'$ appears to be rotated $180°$ from $ABCD$ and displaced. A single $180°$ rotation about the appropriate center point can map one parallelogram onto the other. Since both parallelograms have the same size and shape (congruent), and $A'B'C'D'$ appears to be oriented opposite to $ABCD$, a single rotation of $180°$ 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.
8. Triangle $DEF$ is mapped onto triangle $D'E'F'$ on the coordinate plane by two rigid transformations. Order matters. Which description correctly orders the transformations?
   Answer: Rotate $90^\circ$ clockwise about the origin, then translate up 3 units. 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.
9. Square $PQRS$ is rotated $45°$ counterclockwise about its center to create square $P'Q'R'S'$. Then square $P'Q'R'S'$ is reflected across a line passing through the center of the original square to create square $P''Q''R''S''$. If the final image $P''Q''R''S''$ has the same orientation as the original square $PQRS$, what was the direction of the reflection line?
   Answer: The reflection line was diagonal, making a $22.5°$ angle with the sides of the square After rotating $45°$ counterclockwise, the square's sides are at $45°$ angles to the original orientation. To return the square to its original orientation, the reflection must undo this $45°$ 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 $-45°$ (returning to original position), we need $2θ = 45°$, so $θ = 22.5°$. Therefore, the reflection line makes a $22.5°$ angle with the original sides of the square.

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