Rigid Transformations Worksheet with Answers – Geometry (Free PDF)

Problem 1 Translate a point ★★★

A translation uses the rule (x, y) → (x − 5, y + 4). Find the image of P(3, −2).

Problem 2 Reflect over an axis ★★★

Find the image of Q(−4, 6) after a reflection over (a) the x-axis and (b) the y-axis.

Problem 3 Half-turn about the origin ★★★

Rotate R(7, −3) by 180° about the origin. Find R′.

Problem 4 Quarter-turn rotation ★★★

Rotate S(4, −1) by 90° counterclockwise about the origin. Find S′.

Problem 5 Reflect over y = x ★★★

Reflect T(−2, 5) over the line y = x. Find T′.

Problem 6 Describe a translation ★★★

Triangle ABC has A(−3, 1), B(0, 1) and C(−2, 4). Its image has A′(2, −2). Find the translation vector and the other two image points.

Problem 7 Rigid or not rigid ★★★

Decide whether each rule is a rigid transformation. Test the segment from (0, 0) to (2, 0).

1. (x, y) → (x + 4, y − 1)
2. (x, y) → (2x, 2y)
3. (x, y) → (x, −y)
4. (x, y) → (3x, y)

Problem 8 Reflect a quadrilateral ★★★

Quadrilateral WXYZ has vertices W(2, 1), X(6, 2), Y(7, 5) and Z(3, 4). Find the image after a reflection over the y-axis.

Problem 9 Clockwise rotation ★★★

Triangle DEF has D(2, 1), E(6, 1) and F(6, 4). Rotate it 90° clockwise about the origin and list the image vertices.

Problem 10 Reflect over y = −x ★★★

The rule for a reflection over the line y = −x is (x, y) → (−y, −x). Find the images of P(3, −5) and Q(−2, −6).

Problem 11 Symmetries of a rectangle ★★★

A 6 by 4 rectangle sits at the origin with sides parallel to the axes. Name each reflection and each turn about the origin (less than 360°) that carries it onto itself, then compare with a square.

Problem 12 Regular polygon symmetry ★★★

For a regular pentagon, a regular octagon and a regular 12-gon, find the number of lines of symmetry and the smallest positive angle of rotational symmetry.

Problem 13 Order of two motions ★★★

Point M(−7, 4) is translated by ⟨8, −9⟩ and reflected over the x-axis. Find the final point. Then perform the two steps in the opposite order and compare.

Problem 14 Find the line of reflection ★★★

A reflection maps A(2, 5) to A′(2, −1). Find the line of reflection, then find the image of B(6, −3).

Problem 15 Robot docking route ★★★

A robotics club maps its arena on a grid where one unit is one foot. A robot starts at (−3, 2). It moves by the translation ⟨7, −4⟩, then reflects over the x-axis to reach its dock. Find the dock location and the straight-line length of the first move to the nearest tenth.

Problem 16 Rotation about another center ★★★

Rotate P(5, 3) by 90° counterclockwise about the center C(2, 1). Separately, rotate the original point P by 180° about C. Find both images.

Problem 17 Reflection then translation ★★★

Triangle ABC has A(1, 1), B(4, 1) and C(4, 3). Reflect it over the y-axis, then translate by ⟨0, −5⟩. Find A″, B″ and C″ and verify that AB = A″B″.

Problem 18 Identify the rotation ★★★

Triangle PQR with P(1, 2), Q(4, 2) and R(4, 4) is mapped onto P′(−2, 1), Q′(−2, 4) and R′(−4, 4). Identify the single transformation and write its rule.

Problem 19 Mosaic with four-fold symmetry ★★★

A designer places a tile marker at (3, 1) and turns it by 90° about the origin again and again to build a four-fold mosaic. List the four marker positions. Then name the reflection line that maps (3, 1) onto (1, 3).

Problem 20 Two reflections make a translation ★★★

Reflect P(0, 2) over the line x = 1, then reflect the result over the line x = 4. Find the final point and the single translation that does the same job. Then show it works for a general point (x, y).