section 3: using matrices to transform geometric figures

CHAPTER 4 | MATRICES

Section 3: Using Matrices to Transform Geometric Figures

OBJECTIVES Use matrices to transform a plane

figure.

VOCABULARY translation matrix reflection matrix rotation matrix

WHO USES THIS?

Artists, such as M. C. Escher, may use repeated transformed patterns to create their work. (See Exercise 16.)

4.3: USING MATRICES TO TRANSFORM GEOMETRIC FIGURES

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 TRIABC below are A (-2, -1) , B (0, 3) , and C (1, -2) .

You can also define TRIABC by a matrix:

4.3: USING MATRICES TO TRANSFORM GEOMETRIC FIGURES

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.

EXAMPLE ONE

4.3: USING MATRICES TO TRANSFORM GEOMETRIC FIGURES 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 are proportional.

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

4.3: USING MATRICES TO TRANSFORM GEOMETRIC FIGURES

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 f by the coordinate matrix.

This reverses the x-coordinates and keeps the y-coordinates unchanged.

EXAMPLE 3

4.3: USING MATRICES TO TRANSFORM GEOMETRIC FIGURES

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

EXAMPLE 4

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