Geometry Translations, Reflections and Rotations Worksheet (PDF)

Geometry math worksheet: Translations, Reflections and Rotations – Maths PDF

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This geometry transformations worksheet for Geometry covers translations, reflections and rotations (Common Core G-CO.A.2, G-CO.A.4, G-CO.A.5), with a free PDF to print.

These 20 problems practice translations, reflections, rotations and sequences of rigid motions on the coordinate plane (Common Core G-CO.A.2 to G-CO.A.5). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applied problems with games, cranes, tiles and compositions.

Geometry · Common Core State Standards: G-CO.A.2, G-CO.A.4, G-CO.A.5

Geometry transformations worksheet: exercises 1 to 10

Exercise 1 – Translate a point

Translate \((3, -2)\) by the rule \((x, y) \text{ maps to } (x + 4, y + 5)\).

Exercise 2 – Reflect across the x-axis

Reflect \((5, -3)\) across the x-axis.

Exercise 3 – Reflect across the y-axis

Reflect \((-4, 6)\) across the y-axis.

Exercise 4 – Rotate 90 degrees

Rotate \((2, 5)\) by \(90^{\circ}\) counterclockwise about the origin.

Exercise 5 – Rotate 180 degrees

Rotate \((-3, 4)\) by \(180^{\circ}\) about the origin.

Exercise 6 – Translate a triangle

Triangle ABC.

Triangle ABC has vertices \(A(1, 1)\), \(B(4, 1)\) and \(C(2, 3)\). Translate it by \((x, y) \text{ maps to } (x – 5, y + 2)\) and list the vertices of the image.

Exercise 7 – Reflect a triangle

Triangle DEF.

Reflect triangle DEF, with \(D(-3, 2)\), \(E(-1, 5)\) and \(F(-1, 2)\), across the x-axis. Give the image vertices.

Exercise 8 – Rotate a triangle clockwise

Triangle ABC.

Rotate triangle ABC with \(A(1, 1)\), \(B(4, 1)\) and \(C(2, 3)\) by \(90^{\circ}\) clockwise about the origin. Give the image vertices.

Exercise 9 – Reflect across y = x

Reflect \((3, -7)\) across the line \(y = x\).

Exercise 10 – Reflect across a vertical line

Reflect \((6, 1)\) across the line \(x = 2\).

Exercises 11 and up

Exercise 11 – Find the translation rule

A translation maps \(A(2, 3)\) to \(A^{\prime}(-1, 7)\). Write the rule and find the image of \(B(0, -5)\).

Exercise 12 – Identify the transformation

Segment PQ and its image.

Segment PQ with P(3, 1) and Q(5, 4) maps to segment P′Q′ with P′(-3, 1) and Q′(-5, 4). Name the transformation.

Exercise 13 – Rotate 270 degrees

Rotate \((4, -1)\) by \(270^{\circ}\) counterclockwise about the origin.

Exercise 14 – Reflect, then translate

Reflect \((3, 4)\) across the x-axis, then translate the image by \((x + 2, y – 1)\).

Exercise 15 – Does the order matter?

Translate \((3, 4)\) by \((x + 2, y – 1)\) first and then reflect across the x-axis. Compare with the previous problem.

Exercise 16 – Video game sprite

In a video game, a sprite starts at (6, 2) on the screen grid. It moves 4 units left and 3 units up, and then a mirror effect reflects it across the y-axis. Find its final position.

Exercise 17 – Crane arm

Crane arm at (8, 6).

The tip of a crane arm is at (8, 6) relative to the base at the origin. The arm swings 90° counterclockwise. Find the new tip location and show that the arm length did not change.

Exercise 18 – Floor tile

A parallelogram tile has vertices \((1, 2)\), \((4, 2)\), \((5, 4)\) and \((2, 4)\). Rotate the tile \(180^{\circ}\) about the origin and list the new vertices.

Exercise 19 – Two-step sequence

Starting point.

A point \((2, -3)\) is reflected across the y-axis and then rotated \(90^{\circ}\) counterclockwise about the origin. Find the final point.

Exercise 20 – Rigid motions keep distance

Segment AB has endpoints \(A(1, 2)\) and \(B(4, 6)\). Rotate it \(90^{\circ}\) counterclockwise about the origin and show that the length stays the same.

Key facts behind this geometry transformations worksheet

Key facts

A translation moves every point the same distance in the same direction: (x, y) to (x + a, y + b).
A reflection across the x-axis sends (x, y) to (x, -y); across the y-axis it sends (x, y) to (-x, y).
A 90 degree counterclockwise rotation about the origin sends (x, y) to (-y, x).
Rigid motions preserve distance and angle measure, so the image is congruent to the preimage.
When transformations are combined, the order matters.

How to use this geometry transformations worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the geometry transformations worksheet as a PDF with the button below and keep it with your notes.

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