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Warm Up

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Perform Reflections

Warm Up

Lesson Presentation

Lesson Quiz

1.The y-axis is the perpendicular bisector of AB . If point A is at (–3, 5), what is the location of point B?

(3, 5)

ANSWER

ANSWER

–1

Warm-Up

2.What is the slope of the segment with endpoints at (–7, 4) and (4, –7)?

– 1

–2

6

0

0 1

1

4 7

3.Multiply .

3

2

– 6

–1

7

ANSWER

3

4

Warm-Up

a.

In the linen : x = 3

Point Ais 2 units left of n, so its reflectionA′is 2 units right of nat (5, 3). Also, B′ is 2 units left of nat (1, 2), and C′is 1 unit right of n at (4, 1).

Example 1

The vertices of ABC are A(1, 3),B(5, 2), and C(2, 1). Graph the reflection of ABC described.

SOLUTION

b.

In the linem : y = 1

Point Ais 2 units above m, so A′ is 2 units below mat (1, –1). Also, B′ is 1 unit below mat (5, 0). Because point Cis on line m, you know that C = C′.

Example 1

The vertices of ABC are A(1, 3),B(5, 2), and C(2, 1). Graph the reflection of ABC described.

SOLUTION

ANSWER

Guided Practice

Graph a reflection of ABC from Example 1 in the given line.

1.y = 4

ANSWER

Guided Practice

Graph a reflection of ABC from Example 1 in the given line.

2.x = –3

ANSWER

Guided Practice

Graph a reflection of ABC from Example 1 in the given line.

3.y = 2

The endpoints of FG are F(–1, 2) and G(1, 2). Reflect the segment in the line y = x. Graph the segment and its image.

The slope of y = xis 1. The segment from Fto its image, FF ′ , is perpendicular to the line of reflection y = x, so the slope of FF′ will be –1(because 1(–1) = –1). FromF, move 1.5units right and1.5units down toy = x. From that point, move1.5units right and1.5units down to locateF′(2,–1).

Example 2

SOLUTION

The slope of GG′ will also be –1. From G, move 0.5 units right and 0.5 units down to y = x. Then move 0.5 units right and 0.5 units down to locate G′(2, 1).

Example 2

Reflect FGfrom Example 2 in the line y = –x. Graph FGand its image.

(a, b) (–b, –a)

F(–1, 2)

F ′(–2, 1)

G(1, 2)

G ′(–2, –1)

Example 3

SOLUTION

Use the coordinate rule for reflecting in y = –x.

ANSWER

Guided Practice

4.Graph ABC with vertices A(1, 3), B(4, 4), and C(3, 1). Reflect ABC in the lines y = –x and y = x. Graph each image.

ANSWER

Slope of y = – xis –1.

The slope of FF′ is 1.

The product of their slopes is –1 making them perpendicular.

Guided Practice

′

5.In Example 3, verify that FF is perpendicular to y = –x.

Parking

You are going to buy books. Your friend is going to buy CDs. Where should you park to minimize the distance you both will walk?

Example 4

Reflect Bin line mto obtain B′. Then draw AB′ . Label the intersection of AB′ and mas C. Because AB′ is the shortest distance between Aand B′ and BC = B′C, park at point Cto minimize the combined distance, AC + BC, you both have to walk.

Example 4

SOLUTION

ANSWER

Because A′Bis the shortest distance between A′and Band AC = A′C, park at point Cto minimize the combined distance, AC + BC, you both have to walk.

Guided Practice

6.Look back at Example 4. Answer the question by using a reflection of point Ainstead of pointB.

The vertices of DEFare D(1, 2), E(3, 3), and F(4, 0). Find the reflection of DEFin the y-axis using matrix multiplication. Graph DEFand its image.

Multiply the polygon matrix by the matrix for a reflection in the y-axis.

STEP 1

D E F

–1 0

1 3 4

0 1

2 3 0

Polygon matrix

Reflection matrix

Example 5

SOLUTION

–1(1) + 0(2) –1(3) + 0(3) –1(4) + 0(0)

=

0(1) + 1(2) 0(3) + 1(3) 0(4) + 1(0)

D′ E′ F′

–1 –3 –4

=

2 3 0

Image matrix

Example 5

Example 5

STEP 2

GraphDEFandD′E′F′.

L′(–3, –3), M′(1, –2), and N′(–2, –1)

ANSWER

Guided Practice

The vertices ofLMN areL(–3, 3), M(1, 2),andN(–2, 1).

Find the described reflection using matrix multiplication.

7.Reflect LMNin the x-axis.

L′(3, 3), M′(–1, 2), and N′(2, 1)

ANSWER

Guided Practice

The vertices ofLMN areL(–3, 3), M(1, 2),andN(–2, 1).

Find the described reflection using matrix multiplication.

8.Reflect LMNin the y-axis.

ANSWER

Lesson Quiz

Graph the polygon and its reflection in the given line.

1.Quadrilateral with verticesA(–2, 1), B(1, 4), C(3, 1), D(2, –1) overy = 1.

ANSWER

Lesson Quiz

Graph the polygon and its reflection in the given line.

2.TriangleP(–2, 2), Q(3, 4), R(4, 1) over y = x.

A B C

A’ B’ C’

ANSWER

2 –1 0

– 2 1 0

3 4 –1

3 4 –1

Lesson Quiz

3. Use matrix multiplication to find the Image matrix when ABC is reflected in y-axis.

ANSWER

( , 0)

41

8

Lesson Quiz

4.Find thepointCon thex-axis soAC + BC is a minimum for A(2, 5) andB(7, 3).