Rigid Transformations: Geometry Math Worksheet Answer Key – PDF Download

Problem 1: Translate points

1. Add the components: \((3+(-5),\,-2+4)\) = \((-2,\,2)\).
2. \((-4+7,\,6+(-9))\) = \((3,\,-3)\).

Problem 2: Translate a triangle

Add 3 to each x-coordinate and subtract 5 from each y-coordinate.

\(A^{\prime}(4,\,-4)\), \(B^{\prime}(7,\,-4)\), \(C^{\prime}(5,\,-1)\).

Problem 3: Reflect points over the axes

1. The x-axis reflection changes the sign of y: \((-3,\,-5)\).
2. The y-axis reflection changes the sign of x: \((3,\,5)\).
3. \((4,\,7)\).

Problem 4: Reflect a triangle over the y-axis

Change the sign of each x-coordinate.

\(D^{\prime}(5,\,1)\), \(E^{\prime}(2,\,1)\), \(F^{\prime}(4,\,5)\).

Problem 5: Reflect over y = x

Swap the coordinates.

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

Problem 6: Reflect over a vertical line

The y-coordinate stays the same. The new x-coordinate is \(2(3)-x\).

1. \(2(3)-(-1)=7\): \((7,\,4)\)
2. \(2(3)-5=1\): \((1,\,-2)\)
3. The point is on the line, so it stays: \((3,\,6)\)

Problem 7: Rotate points 90 degrees

Use \((x,\,y)\) becomes \((-y,\,x)\).

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

Problem 8: Rotate 180 degrees and clockwise

1. \((x,\,y)\) becomes \((-x,\,-y)\): \((5,\,-2)\).
2. A \(90^{\circ}\) clockwise turn sends \((x,\,y)\) to \((y,\,-x)\): \((6,\,-2)\).

Problem 9: Rotate a triangle

Use \((x,\,y)\) becomes \((-y,\,x)\) for each vertex.

\(G^{\prime}(-1,\,1)\), \(H^{\prime}(-1,\,5)\), \(J^{\prime}(-3,\,1)\).

Problem 10: Find the translation vector

Compare \(P(-5,\,3)\) with \(P^{\prime}(1,\,0)\): \(1-(-5)=6\) and \(0-3=-3\). The other vertices agree.

The vector is \(\langle 6,\,-3\rangle\).

Problem 11: Find the line of reflection

The line of reflection is the perpendicular bisector of \(\overline{AA^{\prime}}\). \(A(-4,\,3)\) and \(A^{\prime}(-4,\,-1)\) have the same x-coordinate, so the line is horizontal, through the midpoint \((-4,\,\frac{3+(-1)}{2})=(-4,\,1)\).

The line is \(y=1\).

Problem 12: Rigid or not?

1. Image endpoints (4, −1) and (7, 3): length \(\sqrt{3^{2}+4^{2}}=5\). Rigid (a translation).
2. Image endpoints (0, 0) and (6, 8): length 10. Not rigid (a dilation).
3. Image endpoints (0, 0) and (3, −4): length 5. Rigid (a reflection).
4. Image endpoints (0, 0) and (9, 4): length \(\sqrt{81+16}=\sqrt{97}\), about 9.85. Not rigid.
5. Image endpoints (0, 0) and (−4, 3): length 5. Rigid (a rotation of 90° counterclockwise).

Problem 13: Compare distances

1. \(AB=\sqrt{(4-1)^{2}+(6-2)^{2}}=\sqrt{9+16}=5\).
2. \(A^{\prime}(-2,\,4)\) and \(B^{\prime}(1,\,8)\): \(A^{\prime}B^{\prime}=\sqrt{9+16}=5\).
3. \(A^{\prime}(2,\,2)\) and \(B^{\prime}(8,\,6)\): \(A^{\prime}B^{\prime}=\sqrt{36+16}=\sqrt{52}\), about 7.21.

The translation keeps length 5, so it is rigid. The stretch changes it, so it is not rigid.

Problem 14: Symmetry of a rectangle

1. One line joins the midpoints of the long sides and one joins the midpoints of the short sides, so there are 2 lines. The diagonals are not lines of symmetry because the rectangle is not a square.
2. The rectangle maps onto itself after a turn of \(180^{\circ}\) and a full turn of \(360^{\circ}\).

Problem 15: Symmetry of a regular hexagon

A regular polygon with \(n\) sides has \(n\) lines of symmetry, so a hexagon has 6 lines.

The smallest turn is \(360 : 6=60\) degrees. The rotations are \(60^{\circ}\), \(120^{\circ}\), \(180^{\circ}\), \(240^{\circ}\) and \(300^{\circ}\).

Problem 16: Symmetries of other shapes

1. 0 lines. It maps onto itself after a rotation of \(180^{\circ}\) about its center.
2. 1 line (through the midpoints of the parallel sides). It has no rotational symmetry other than a full turn.
3. 5 lines. The smallest rotation is \(360 : 5=72\), so \(72^{\circ}\), \(144^{\circ}\), \(216^{\circ}\) and \(288^{\circ}\).

Problem 17: Translate, then reflect

1. \((2,\,3)\) becomes \((-2,\,4)\) after the translation, then \((-2,\,-4)\) after the reflection.
2. \((0,\,-2)\) becomes \((-4,\,-1)\), then \((-4,\,1)\).

Problem 18: Rotate, then reflect

1. After the rotation: \(K^{\prime}(-1,\,1)\), \(L^{\prime}(-1,\,3)\), \(M^{\prime}(-4,\,1)\). After the reflection over the y-axis: \(K^{\prime\prime}(1,\,1)\), \(L^{\prime\prime}(1,\,3)\), \(M^{\prime\prime}(4,\,1)\).
2. Each final point is the original with its coordinates swapped, for example \(L(3,\,1)\) to \(L^{\prime\prime}(1,\,3)\). The single transformation is a reflection over the line \(y=x\).

Problem 19: Describe a sequence

The orientation of the triangle is flipped, so use a reflection. One answer: reflect over the x-axis, then translate by \(\langle 7,\,-2\rangle\).

Reflection: \(A^{\prime}(-5,\,-1)\), \(B^{\prime}(-2,\,-1)\), \(C^{\prime}(-5,\,-3)\). Translation: \(A^{\prime\prime}(2,\,-3)\), \(B^{\prime\prime}(5,\,-3)\), \(C^{\prime\prime}(2,\,-5)\), which matches the final image.

Problem 20: Video game character

Translation: \((3-7,\,-4+6)\) = \((-4,\,2)\).

Rotation: \((x,\,y)\) becomes \((-y,\,x)\), so \((-4,\,2)\) becomes \((-2,\,-4)\).

Problem 21: Park statues

The y-coordinate stays 3. The new x-coordinate is \(2(1)-(-6)=8\).

The second statue is at \((8,\,3)\). Check: the walkway \(x=1\) is halfway between \(-6\) and \(8\).

Problem 22: Logo design

A \(180^{\circ}\) rotation sends \((x,\,y)\) to \((-x,\,-y)\).

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

Problem 23: Pinwheel

The blades are equally spaced, so the smallest turn is \(360 : 8=45\) degrees.

All rotations are multiples of \(45^{\circ}\): \(45^{\circ}\), \(90^{\circ}\), \(135^{\circ}\), \(180^{\circ}\), \(225^{\circ}\), \(270^{\circ}\), \(315^{\circ}\), \(360^{\circ}\).

Problem 24: Perimeter after a sequence

Translation: \(A^{\prime}(2,\,-2)\), \(B^{\prime}(5,\,-2)\), \(C^{\prime}(2,\,2)\). Reflection: \(A^{\prime\prime}(2,\,2)\), \(B^{\prime\prime}(5,\,2)\), \(C^{\prime\prime}(2,\,-2)\).

Original side lengths: \(AB=3\), \(AC=4\), \(BC=\sqrt{9+16}=5\), perimeter \(12\). The image sides are 3, 4 and 5 as well, so the perimeter is 12 both times . Rigid transformations preserve length.

Problem 25: Does order matter?

Translate first: \((1,\,2)\) becomes \((4,\,2)\), then \((-4,\,2)\).

Reflect first: \((1,\,2)\) becomes \((-1,\,2)\), then \((2,\,2)\).

The results differ, so Priya is wrong: the order of a sequence matters.

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