Rigid Transformations: Geometry Math Test Answer Key – PDF Download

Question 1: Translate a triangle (2 points)

Subtract 4 from each x-coordinate and add 3 to each y-coordinate.

\(A^{\prime}(-2,\,0)\), \(B^{\prime}(2,\,0)\), \(C^{\prime}(-1,\,2)\).

Scoring: 2 points if all three vertices are correct; 1 point if two are correct.

Question 2: Reflect over y = x (2 points)

Reflection over \(y=x\) swaps the coordinates of each point.

\(J^{\prime}(-4,\,1)\), \(K^{\prime}(-1,\,1)\), \(L^{\prime}(-4,\,4)\).

Scoring: 2 points if all three are correct; 1 point if two are correct.

Question 3: Rotate clockwise (2 points)

A \(90^{\circ}\) clockwise rotation sends \((x,\,y)\) to \((y,\,-x)\).

1. \((2,\,-3)\)
2. \((5,\,4)\)
3. \((-6,\,0)\)

Scoring: 1 point for two correct images; 2 points for all three.

Question 4: Sequence of transformations (3 points)

1. Reflection over the y-axis: \(D^{\prime}(-1,\,2)\), \(E^{\prime}(-4,\,2)\), \(F^{\prime}(-1,\,6)\). Translation: \(D^{\prime\prime}(1,\,-1)\), \(E^{\prime\prime}(-2,\,-1)\), \(F^{\prime\prime}(1,\,3)\).
2. \(DE=4-1=3\) and \(D^{\prime\prime}E^{\prime\prime}=1-(-2)=3\). The lengths are equal, as expected for rigid transformations.

Scoring: Part (a): 1 point for the correct intermediate image, 1 point for the final image. Part (b): 1 point for both lengths.

Question 5: Identify the transformation (2 points)

Each point \((x,\,y)\) goes to \((-x,\,-y)\), for example \(A(1,\,1)\) to \(A^{\prime}(-1,\,-1)\).

The transformation is a rotation of \(180^{\circ}\) about the origin.

Scoring: 1 point for naming a rotation (a reflection over both axes in turn is also accepted); 1 point for the angle \(180^{\circ}\) and the center at the origin.

Question 6: Rigid or not rigid (2 points)

1. \(AB=4-1=3\). The image endpoints are \(A^{\prime}(1,\,3)\) and \(B^{\prime}(1,\,12)\), so \(A^{\prime}B^{\prime}=12-3=9\).
2. No. The length changed from 3 to 9, and rigid transformations preserve length.

Scoring: 1 point for both lengths; 1 point for the conclusion with the reason.

Question 7: Symmetry (3 points)

1. 8 lines: a regular polygon with \(n\) sides has \(n\) lines.
2. \(360 : 8=45\): \(45^{\circ}\).
3. 2 lines, one through the midpoints of each pair of opposite sides.

Scoring: 1 point per correct part.

Question 8: Describe a sequence (2 points)

One answer: rotate \(180^{\circ}\) about the origin, then translate by \(\langle -1,\,4\rangle\).

Rotation: \(A^{\prime}(6,\,-1)\), \(B^{\prime}(2,\,-1)\), \(C^{\prime}(2,\,-3)\), \(D^{\prime}(6,\,-3)\). Translation: \(A^{\prime\prime}(5,\,3)\), \(B^{\prime\prime}(1,\,3)\), \(C^{\prime\prime}(1,\,1)\), \(D^{\prime\prime}(5,\,1)\), which match the image.

Scoring: 1 point for a correct first transformation; 1 point for a second transformation that completes the mapping. Other valid sequences earn full credit.

Question 9: Robot position (2 points)

Reflection over the x-axis: \((-3,\,4)\) becomes \((-3,\,-4)\).

Translation: \((-3+5,\,-4+2)\) = \((2,\,-2)\).

Scoring: 1 point for the reflected point; 1 point for the final position.

⬇ Download the Geometry Rigid Transformations test answer key (PDF, 105 KB)

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