Translations, Reflections and Rotations: Grade 8 Math Worksheet Answer Key – PDF Download

Grade 8 Transformations Worksheet Answer Key: problems 1–5

Problem 1: Translate a point

Add 4 to \(x\) and 5 to \(y\): \((3+4,\,-2+5)=(7,3)\).

The image is \((7,3)\).

Problem 2: Reflect a point in the x-axis

The rule is \((x,y)\to(x,-y)\). So \((5,-3)\to(5,3)\).

The image is \((5,3)\).

Problem 3: Reflect a point in the y-axis

The rule is \((x,y)\to(-x,y)\). So \((-4,6)\to(4,6)\).

The image is \((4,6)\).

Problem 4: Rotate a point 180 degrees

The rule is \((x,y)\to(-x,-y)\). So \((2,-7)\to(-2,7)\).

The image is \((-2,7)\).

Problem 5: Rotate a point 90 degrees

The rule is \((x,y)\to(-y,x)\). So \((3,5)\to(-5,3)\).

The image is \((-5,3)\).

Grade 8 Transformations Worksheet Answer Key: problems 6–10

Problem 6: Rotate a point 270 degrees

The rule is \((x,y)\to(y,-x)\). So \((4,-1)\to(-1,-4)\).

The image is \((-1,-4)\).

Problem 7: Translate a triangle

Add 3 to each \(x\) and subtract 4 from each \(y\).

\(J^{\prime}(-1,-2)\), \(K^{\prime}(2,-2)\), \(L^{\prime}(-1,1)\).

The image is triangle \(J^{\prime}K^{\prime}L^{\prime}\) as listed.

Problem 8: Reflect a triangle

Change the sign of each \(x\)-coordinate: \((x,y)\to(-x,y)\).

\(P^{\prime}(-1,1)\), \(Q^{\prime}(-4,2)\), \(R^{\prime}(-2,5)\).

Problem 9: Reflect a trapezoid

Change the sign of each \(y\)-coordinate: \((x,y)\to(x,-y)\).

\(A^{\prime}(-5,-2)\), \(B^{\prime}(-1,-2)\), \(C^{\prime}(-2,-5)\), \(D^{\prime}(-4,-5)\).

The top side of length 3 and the bottom side of length 4 keep their lengths, so the image is congruent.

Problem 10: Rotate a triangle 90 degrees

Use \((x,y)\to(-y,x)\) on each vertex.

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

Grade 8 Transformations Worksheet Answer Key: problems 11–15

Problem 11: Rotate a parallelogram

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

\(W^{\prime}(-1,-1)\), \(X^{\prime}(-5,-1)\), \(Y^{\prime}(-6,-3)\), \(Z^{\prime}(-2,-3)\).

Opposite sides of the image are still parallel.

Problem 12: Rotate a triangle 270 degrees

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

\(G^{\prime}(1,3)\), \(H^{\prime}(4,3)\), \(I^{\prime}(1,1)\).

Problem 13: Describe a translation

For \(A\): \(x\) changes from \(-2\) to \(3\), which is \(+5\). \(y\) changes from \(1\) to \(-3\), which is \(-4\).

Check \(B\): \((0+5,\,4-4)=(5,0)\).

The translation is 5 units right and 4 units down, \((x,y)\to(x+5,y-4)\).

Problem 14: Identify the reflection

\(A(2,1)\) goes to \(A^{\prime}(2,-1)\): the \(x\)-coordinate stays and the \(y\)-coordinate changes sign. \(C(2,3)\) goes to \((2,-3)\), which agrees.

It is a reflection across the \(x\)-axis, \((x,y)\to(x,-y)\).

Problem 15: Identify the rotation

Test \(A(1,2)\) with the three rules. \(90^{\circ}\) gives \((-2,1)\). \(180^{\circ}\) gives \((-1,-2)\). \(270^{\circ}\) gives \((2,-1)\), which matches \(A^{\prime}\).

Check \(B(4,2)\): \((y,-x)=(2,-4)\).

The rotation is \(270^{\circ}\) counterclockwise.

Grade 8 Transformations Worksheet Answer Key: problems 16–20

Problem 16: Write the rule

In both pairs the \(y\)-coordinate is unchanged and the \(x\)-coordinate changes sign.

It is a reflection across the \(y\)-axis, \((x,y)\to(-x,y)\).

Problem 17: What stays the same

A reflection is a rigid motion, so it keeps segment lengths and angle measures.

\(D^{\prime}E^{\prime}=6\text{ cm}\), \(E^{\prime}F^{\prime}=8\text{ cm}\), and \(\angle E^{\prime}=90^{\circ}\).

Problem 18: Parallel lines

Images: \(p^{\prime}\) joins \((-1,-1)\) and \((-3,-5)\); \(q^{\prime}\) joins \((-3,-1)\) and \((-5,-5)\).

Slope of \(p^{\prime}\): \(\frac{-5-(-1)}{-3-(-1)}=\frac{-4}{-2}=2\). Slope of \(q^{\prime}\): \(\frac{-5-(-1)}{-5-(-3)}=\frac{-4}{-2}=2\).

The slopes are equal, so the images are parallel.

Problem 19: Translate, then reflect

Step 1, \((x-2,\,y-3)\): \(A^{\prime}(-1,-1)\), \(B^{\prime}(1,-1)\), \(C^{\prime}(-1,2)\).

Step 2, \((x,y)\to(-x,y)\): \(A^{\prime}^{\prime}\)=\((1,-1)\), \(B^{\prime}^{\prime}\)=\((-1,-1)\), \(C^{\prime}^{\prime}\)=\((1,2)\).

The final image has vertices \((1,-1)\), \((-1,-1)\), \((1,2)\).

Problem 20: Rotate, then reflect

Step 1, \((-y,x)\): \((-1,1)\), \((-1,4)\), \((-3,4)\).

Step 2, \((x,-y)\): \((-1,-1)\), \((-1,-4)\), \((-3,-4)\).

The final image has vertices \((-1,-1)\), \((-1,-4)\), \((-3,-4)\).

Grade 8 Transformations Worksheet Answer Key: problems 21–25

Problem 21: Park map

Reflect across the \(y\)-axis: \((-6,2)\to(6,2)\).

The benches have the same \(y\)-coordinate, so the distance is \(6-(-6)=12\) units.

The second bench is at \((6,2)\), and the benches are 12 yards apart.

Problem 22: Game piece

First card: \((3,2)\to(-2,3)\). Second card: \((-2,3)\to(-3,-2)\).

Directly, \(180^{\circ}\) gives \((-3,-2)\).

The piece is at \((-3,-2)\), the same as one \(180^{\circ}\) rotation.

Problem 23: Wallpaper stamp

The 4th stamp is 3 translations from the first, so it has moved \(3\times 4=12\) units right.

\((1+12,\,1)=(13,1)\).

The vertex is at \((13,1)\).

Problem 24: Find the original point

The rule is \((x,y)\to(y,-x)\). So \(y=-3\) and \(-x=8\), which gives \(x=-8\).

Check: \((-8,-3)\to(-3,8)\).

The original point is \(Q(-8,-3)\).

Problem 25: Two reflections

First: \((3,2)\to(3,-2)\). Second: \((3,-2)\to(-3,-2)\).

A \(180^{\circ}\) rotation sends \((3,2)\) to \((-3,-2)\) too.

The result is \((-3,-2)\), equal to a \(180^{\circ}\) rotation.

Grade 8 Transformations Worksheet Answer Key: problems 26–26

Problem 26: Legs and the right angle

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

Original legs: \(AB=5-1=4\) and \(BC=4-1=3\).

Image legs: \(A^{\prime}B^{\prime}=5-1=4\) (vertical) and \(B^{\prime}C^{\prime}=-1-(-4)=3\) (horizontal).

A vertical and a horizontal segment meet at \(90^{\circ}\), so the right angle and both lengths are preserved.

How to use this grade 8 transformations worksheet answer key

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