1. 16 Using Matrices to Transform Geometric Figures Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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

3. Use matrices to transform a plane figure. Objective

4. translation matrix reflection matrix rotation matrix Vocabulary

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), & C'(–3, 1).

10. YOUR TURN! 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), & 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, & 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. Your Turn! 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. 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. YOUR TURN! 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. Your Turn! 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. 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. The image A'B'C'D' is rotated 90° counterclockwise. The image A''B''C''D'' is rotated 90° clockwise. A. B.

25. Example 4 Continued

26. Your Turn! 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. Your Turn! 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. 3. 4.