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9.3
Warm UpWarm Up
Lesson QuizLesson Quiz
Lesson PresentationLesson Presentation
Perform Reflections
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9.3 Warm-Up
(3, 5)ANSWER
2. What is the slope of the segment with endpoints at (–7, 4) and (4, –7)?
ANSWER –1
1. The y-axis is the perpendicular bisector of AB . If point A is at (–3, 5), what is the location of point B?
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9.3 Warm-Up
3. Multiply . 00 1
– 1 14 7
–263
ANSWER –1 7
2– 63 4
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9.3 Example 1
SOLUTION
The vertices of ABC are A(1, 3), B(5, 2), and C(2, 1). Graph the reflection of ABC described.
a. In the line n : x = 3
Point A is 2 units left of n, so its reflection A′ is 2 units right of n at (5, 3). Also, B′ is 2 units left of n at (1, 2), and C′ is 1 unit right of n at (4, 1).
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9.3 Example 1
SOLUTION
The vertices of ABC are A(1, 3), B(5, 2), and C(2, 1). Graph the reflection of ABC described.
b. In the line m : y = 1
Point A is 2 units above m, so A′ is 2 units below m at (1, –1). Also, B′ is 1 unit below m at (5, 0). Because point C is on line m, you know that C = C′.
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9.3 Guided Practice
Graph a reflection of ABC from Example 1 in the given line.
1. y = 4
ANSWER
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9.3 Guided Practice
Graph a reflection of ABC from Example 1 in the given line.
2. x = –3
ANSWER
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9.3 Guided Practice
Graph a reflection of ABC from Example 1 in the given line.
3. y = 2
ANSWER
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9.3 Example 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.
SOLUTION
The slope of y = x is 1. The segment from F to its image, FF ′ , is perpendicular to the line of reflection y = x, so the slope of FF′ will be –1 (because 1(–1) = –1). From F, move 1.5 units right and 1.5 units down to y = x. From that point, move 1.5 units right and 1.5 units down to locate F′(2,–1).
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9.3 Example 2
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).
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9.3 Example 3
SOLUTION
Use the coordinate rule for reflecting in y = –x.
(a, b) (–b, –a)
G(1, 2) G ′(–2, –1)
F(–1, 2) F ′(–2, 1)
Reflect FG from Example 2 in the line y = –x. Graph FG and its image.
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9.3 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
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9.3 Guided Practice
5. In Example 3, verify that FF is perpendicular to y = –x.′
Slope of y = – x is –1.
The slope of FF′ is 1.
The product of their slopes is –1 making them perpendicular.
ANSWER
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9.3 Example 4
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?
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9.3 Example 4
SOLUTION
Reflect B in line m to obtain B′. Then draw AB′ . Label the intersection of AB′ and m as C. Because AB′ is the shortest distance between A and B′ and BC = B′C, park at point C to minimize the combined distance, AC + BC, you both have to walk.
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9.3 Guided Practice
6. Look back at Example 4. Answer the question by using a reflection of point A instead of point B.
ANSWER
Because A′B is the shortest distance between A′ and B and AC = A′C, park at point C to minimize the combined distance, AC + BC, you both have to walk.
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9.3 Example 5
SOLUTION
The vertices of DEF are D(1, 2), E(3, 3), and F(4, 0). Find the reflection of DEF in the y-axis using matrix multiplication. Graph DEF and its image.
STEP 1
Multiply the polygon matrix by the matrix for a reflection in the y-axis.
–1 0
0 1
Reflection matrix
1 3 4
2 3 0
Polygon
matrix
D E F
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9.3 Example 5
–1(1) + 0(2) –1(3) + 0(3) –1(4) + 0(0)=
0(1) + 1(2) 0(3) + 1(3) 0(4) + 1(0)
–1 –3 –4
2 3 0=
Image matrix
D′ E′ F′
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9.3 Example 5
STEP 2
Graph DEF and D′E′F′.
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9.3 Guided Practice
The vertices of LMN are L(–3, 3), M(1, 2), and N(–2, 1).
Find the described reflection using matrix multiplication.
7. Reflect LMN in the x-axis.
ANSWER L′(–3, –3), M′(1, –2), and N′(–2, –1)
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9.3 Guided Practice
The vertices of LMN are L(–3, 3), M(1, 2), and N(–2, 1).
Find the described reflection using matrix multiplication.
8. Reflect LMN in the y-axis.
ANSWER L′(3, 3), M′(–1, 2), and N′(2, 1)
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9.3 Lesson Quiz
1. Quadrilateral with vertices A(–2, 1), B(1, 4), C(3, 1), D(2, –1) over y = 1.
ANSWER
Graph the polygon and its reflection in the given line.
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9.3 Lesson Quiz
Graph the polygon and its reflection in the given line.
ANSWER
2. Triangle P(–2, 2), Q(3, 4), R(4, 1) over y = x.
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9.3 Lesson Quiz
3. Use matrix multiplication to find the Image matrix when ABC is reflected in y-axis.
2 –1 0 3 4 –1
A B C
– 2 1 03 4 –1
A’ B’ C’ANSWER
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9.3 Lesson Quiz
4. Find the point C on the x-axis so AC + BC is a minimum for A(2, 5) and B(7, 3).
41 8( , 0)ANSWER