1 / 29

# 4-3 - PowerPoint PPT Presentation

Using Matrices to Transform Geometric Figures. 4-3. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Perform the indicated operation. 1. 2. 3. Objective. Use matrices to transform a plane figure. Vocabulary. t ranslation m atrix r eflection m atrix

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' 4-3' - holland

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Geometric Figures

4-3

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2

Perform the indicated operation.

1.

2.

3.

Use matrices to transform a plane figure.

translation matrix

reflection matrix

rotation matrix

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 on a coordinate plane by naming the ordered pairs that define its vertices.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.

Reading Math on a coordinate plane by naming the ordered pairs that define its vertices.

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.

Example 1: on a coordinate plane by naming the ordered pairs that define its vertices.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 on a coordinate plane by naming the ordered pairs that define its vertices.

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

Check It Out! on a coordinate plane by naming the ordered pairs that define its vertices. 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! on a coordinate plane by naming the ordered pairs that define its vertices. Example 1 Continued

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

A on a coordinate plane by naming the ordered pairs that define its vertices.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: Using Matrices to Enlarge a Figure on a coordinate plane by naming the ordered pairs that define its vertices.

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 on a coordinate plane by naming the ordered pairs that define its vertices.

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

Enlarge on a coordinate plane by naming the ordered pairs that define its vertices.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

D on a coordinate plane by naming the ordered pairs that define its vertices.'E'F', the image of DEF, has coordinates

Check It Out! Example 2 Continued

A on a coordinate plane by naming the ordered pairs that define its vertices.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.

Caution on a coordinate plane by naming the ordered pairs that define its vertices.

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

Example 3: Using Matrices to Reflect a Figure on a coordinate plane by naming the ordered pairs that define its vertices.

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 on a coordinate plane by naming the ordered pairs that define its vertices.

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

To reflect a figure across the on a coordinate plane by naming the ordered pairs that define its vertices.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

Check It Out! on a coordinate plane by naming the ordered pairs that define its vertices. Example 3

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

A on a coordinate plane by naming the ordered pairs that define its vertices.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 on a coordinate plane by naming the ordered pairs that define its vertices.

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.

Example 4 Continued on a coordinate plane by naming the ordered pairs that define its vertices.

Check It Out! on a coordinate plane by naming the ordered pairs that define its vertices. 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! on a coordinate plane by naming the ordered pairs that define its vertices. Example 4 Continued

Lesson Quiz on a coordinate plane by naming the ordered pairs that define its vertices.

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 on a coordinate plane by naming the ordered pairs that define its vertices.

1.

2.

3. 4.