Geometry: Transformations Worksheet

Question 1

Triangle ABC has vertices A(2, 1), B(5, 1), and C(3, 4). Translate the triangle 4 units left and 3 units up. Give the coordinates of A', B', and C'.

Accepted Answer

The translation rule is (x, y) to (x - 4, y + 3). The image coordinates are A'(-2, 4), B'(1, 4), and C'(-1, 7).

Question 2

Point P(-6, 4) is reflected across the y-axis. What are the coordinates of P'?

Accepted Answer

A reflection across the y-axis changes the sign of the x-coordinate and keeps the y-coordinate the same. Therefore, P' is (6, 4).

Question 3

Point M(3, -7) is reflected across the x-axis. What are the coordinates of M'?

Accepted Answer

A reflection across the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate. Therefore, M' is (3, 7).

Question 4

Quadrilateral WXYZ has vertices W(-2, 1), X(1, 3), Y(4, 0), and Z(0, -2). Rotate the quadrilateral 90 degrees counterclockwise about the origin. Give the coordinates of W', X', Y', and Z'.

Accepted Answer

A 90 degree counterclockwise rotation about the origin uses the rule (x, y) to (-y, x). The image coordinates are W'(-1, -2), X'(-3, 1), Y'(0, 4), and Z'(2, 0).

Question 5

Point R(-5, 2) is rotated 180 degrees about the origin. What are the coordinates of R'?

Accepted Answer

A 180 degree rotation about the origin changes the sign of both coordinates. Therefore, R' is (5, -2).

Question 6

Triangle DEF has vertices D(1, 2), E(4, 2), and F(2, 6). Dilate the triangle about the origin by a scale factor of 3. Give the coordinates of D', E', and F'.

Accepted Answer

A dilation about the origin with scale factor 3 multiplies each coordinate by 3. The image coordinates are D'(3, 6), E'(12, 6), and F'(6, 18).

Question 7

A figure is transformed by the rule (x, y) to (x + 7, y - 2). Name the transformation and describe it in words.

Accepted Answer

The transformation is a translation. It moves every point 7 units to the right and 2 units down.

Question 8

A point (x, y) is transformed to (-x, y). Name the transformation and identify the line of reflection.

Accepted Answer

The transformation is a reflection across the y-axis because the x-coordinate changes sign while the y-coordinate stays the same.

Question 9

Segment AB has endpoints A(-1, 5) and B(3, 2). It is translated to A'(4, 1) and B'(8, -2). What translation vector was used?

Accepted Answer

From A to A', the change is 5 units right and 4 units down. From B to B', the same change occurs. The translation vector is <5, -4>.

Question 10

Triangle JKL has vertices J(2, -1), K(6, -1), and L(4, -4). First reflect the triangle across the x-axis, then translate it 3 units left. Give the final coordinates of J'', K'', and L''.

Accepted Answer

After reflecting across the x-axis, the coordinates are J'(2, 1), K'(6, 1), and L'(4, 4). Translating 3 units left gives J''(-1, 1), K''(3, 1), and L''(1, 4).

Question 11

A triangle has side lengths 4, 7, and 8. It is dilated by a scale factor of 1/2. What are the side lengths of the image triangle?

Accepted Answer

A dilation multiplies all lengths by the scale factor. The image side lengths are 2, 3.5, and 4.

Question 12

Rectangle ABCD has vertices A(0, 0), B(6, 0), C(6, 4), and D(0, 4). It is dilated about the origin by a scale factor of 2. What is the perimeter of the image rectangle?

Accepted Answer

The original rectangle has side lengths 6 and 4, so its perimeter is 20. A dilation by a scale factor of 2 doubles all lengths, so the image perimeter is 40 units.

Question 13

A figure is rotated 90 degrees clockwise about the origin. Write the coordinate rule for this transformation and use it to find the image of Q(-3, 8).

Accepted Answer

A 90 degree clockwise rotation about the origin uses the rule (x, y) to (y, -x). For Q(-3, 8), the image is Q'(8, 3).

Question 14

Two triangles are congruent. Triangle A is mapped onto Triangle B by a single transformation. The orientation is reversed, and corresponding points are the same distance from the y-axis on opposite sides. What transformation maps Triangle A to Triangle B?

Accepted Answer

The transformation is a reflection across the y-axis. A reflection reverses orientation and places corresponding points the same distance from the line of reflection.

Question 15

Point S(4, -2) is transformed by the rule (x, y) to (2x - 1, 2y + 3). Find S' and describe the transformation as a dilation followed by a translation.

Accepted Answer

Substituting S(4, -2) gives S'(2(4) - 1, 2(-2) + 3), so S'(7, -1). This can be described as a dilation about the origin by a scale factor of 2 followed by a translation 1 unit left and 3 units up.

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Geometry: Transformations

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