1. 4-3 Warm Up Lesson Presentation Lesson Quiz
Using Matrices to Transform
Geometric Figures
4-3
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. Warm Up
Perform the indicated operation. 1. 2. 3.

3. Objective
Use matrices to transform a plane figure.

4. Vocabulary
translation matrix
reflection matrix
rotation matrix

5. 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

6. 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.

7. 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

8. 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

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

10. 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

11. 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).

12. 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.

13. 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

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

15. 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 .

16. Check It Out! Example 2 Continued
D'E'F', the image of DEF, has coordinates

17. 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.

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

19. 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.

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

21. 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. .

22. Check It Out! Example 3
The coordinates of the vertices of the image are J'(3, –4), K'(4, –2), L'(1, 2).

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

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

25. Example 4 Continued

26. 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°.

27. Check It Out! Example 4 Continued

28. 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.

29. Lesson Quiz 1. 2.