Using Matrices to Transform Geometric Figures

Warm Up

Lesson Presentation

Lesson Quiz

Warm UpPerform the indicated operation.

1.

2.

3.

Use matrices to transform a plane figure.

Objective

translation matrixreflection matrixrotation matrix

Vocabulary

You can describe the position, shape, and size of a polygon on a coordinate plane by naming the ordered pairs that define its vertices.

The coordinates of ΔABC below are A (–2, –1), B (0, 3), and C (1, –2) .

You can also define ΔABC by a matrix:

x-coordinates y-coordinates

A translation matrix is a matrix used to translate coordinates on the coordinate plane. The matrix sum of a preimage and a translation matrix gives the coordinates of the translated image.

The prefix pre- means “before,” so the preimage is the original figure before any transformations are applied. The image is the resulting figure after a transformation.

Reading Math

Example 1: Using Matrices to Translate a Figure

Translate ΔABC with coordinates A(–2, 1), B(3, 2), and C(0, –3), 3 units left and 4 units up. Find the coordinates of the vertices of the image, and graph.

The translation matrix will have –3 in all entries in row 1 and 4 in all entries in row 2.

x-coordinates y-coordinates

Example 1 Continued

A'B'C', the image of ABC, has coordinates A'(–5, 5), B'(0, 6), and C'(–3, 1).

Check It Out! Example 1

Translate ΔGHJ with coordinates G(2, 4), H(3, 1), and J(1, –1) 3 units right and 1 unit down. Find the coordinates of the vertices of the image and graph.

The translation matrix will have 3 in all entries in row 1 and –1 in all entries in row 2.

x-coordinates y-coordinates

Check It Out! Example 1 Continued

G'H'J', the image of GHJ, has coordinates G'(5, 3), H'(6, 0), and J'(4, –2).

A dilation is a transformation that scales—enlarges or reduces—the preimage, resulting in similar figures. Remember that for similar figures, the shape is the same but the size may be different. Angles are congruent, and side lengths areproportional.

When the center of dilation is the origin, multiplying the coordinate matrix by a scalar gives the coordinates of the dilated image. In this lesson, all dilations assume that the origin is the center of dilation.

Example 2: Using Matrices to Enlarge a Figure

Enlarge ΔABC with coordinates A(2, 3), B(1, –2), and C(–3, 1), by a factor of 2. Find the coordinates of the vertices of the image, and graph.

Multiply each coordinate by 2 by multiplying each entry by 2.

x-coordinates y-coordinates

Example 2 Continued

A'B'C', the image of ABC, has coordinates A'(4, 6), B'(2, –4), and C'(–6, 2).

Check It Out! Example 2

Enlarge ΔDEF with coordinates D(2, 3), E(5, 1), and F(–2, –7) a factor of . Find the coordinates of the vertices of the image, and graph.

Multiply each coordinate by by multiplying each

entry by .

Check It Out! Example 2 Continued

D'E'F', the image of DEF, has coordinates

A reflection matrix is a matrix that creates a mirror image by reflecting each vertex over a specified line of symmetry. To reflect a figure across the y-axis, multiply

by the coordinate matrix. This reverses the x-coordinates and keeps the y-coordinates unchanged.

Matrix multiplication is not commutative. So be sure to keep the transformation matrix on the left!

Caution

Example 3: Using Matrices to Reflect a Figure

Reflect ΔPQR with coordinates P(2, 2), Q(2, –1), and R(4, 3) across the y-axis. Find the coordinates of the vertices of the image, and graph.

Each x-coordinate is multiplied by –1.

Each y-coordinate is multiplied by 1.

Example 3 Continued

The coordinates of the vertices of the image are P'(–2, 2), Q'(–2, –1), and R'(–4, 3).

Check It Out! Example 3

To reflect a figure across the x-axis, multiply by

Reflect ΔJKL with coordinates J(3, 4), K(4, 2), and L(1, –2) across the x-axis. Find the coordinates of the vertices of the image and graph.

.

Check It Out! Example 3

The coordinates of the vertices of the image are J'(3, –4), K'(4, –2), L'(1, 2).

A rotation matrix is a matrix used to rotate a figure. Example 4 gives several types of rotation matrices.

Example 4: Using Matrices to Rotate a Figure

Use each matrix to rotate polygon ABCD with coordinates A(0, 1), B(2, –4), C(5, 1), and D(2, 3) about the origin. Graph and describe the image.

The image A'B'C'D' is rotated 90° counterclockwise.

The image A''B''C''D'' is rotated 90° clockwise.

A.

B.

Example 4 Continued

Check It Out! Example 4

Use

Rotate ΔABC with coordinates A(0, 0), B(4, 0), and C(0, –3) about the origin. Graph and describe the image.

A'(0, 0), B'(-4, 0), C'(0, 3); the image is rotated 180°.

Check It Out! Example 4 Continued

Lesson Quiz

Transform triangle PQR with vertices P(–1, –1), Q(3, 1), R(0, 3). For each, show the matrix transformation and state the vertices of the image.

1. Translation 3 units to the left and 2 units up.

2. Dilation by a factor of 1.5.

3. Reflection across the x-axis.

4. 90° rotation, clockwise.

Lesson Quiz

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