 # 4.2 Refl ections Essential Question ANALYZING

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4.2 Refl ections Essential Question ANALYZING
```4.2
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
G.3.A
G.3.B
G.3.C
G.3.D
Reflections
Essential Question
How can you reflect a figure in a
coordinate plane?
Reflecting a Triangle Using a Reflective Device
Work with a partner. Use a straightedge to draw any triangle on paper. Label
it △ABC.
a. Use the straightedge to draw a line that does not pass through the triangle.
Label it m.
b. Place a reflective device on line m.
c. Use the reflective device to plot the images of the vertices of △ABC. Label the
images of vertices A, B, and C as A′, B′, and C′, respectively.
d. Use a straightedge to draw △A′B′C′ by connecting the vertices.
ANALYZING
MATHEMATICAL
RELATIONSHIPS
To be proficient in math,
you need to look closely
to discern a pattern
or structure.
Reflecting a Triangle in a Coordinate Plane
Work with a partner. Use dynamic geometry software to draw any triangle and label
it △ABC.
a. Reflect △ABC in the y-axis to form △A′B′C′.
b. What is the relationship between the coordinates of the vertices of △ABC and
those of △A′B′C′?
c. What do you observe about the side lengths and angle measures of the two triangles?
d. Reflect △ABC in the x-axis to form △A′B′C′. Then repeat parts (b) and (c).
Sample
C
A
C′
4
A′
3
2
1
0
−3
−2
B
−1
0
−1
1
2
3
4
B′
Points
A(−3, 3)
B(−2, −1)
C(−1, 4)
Segments
AB = 4.12
BC = 5.10
AC = 2.24
Angles
m∠A = 102.53°
m∠B = 25.35°
m∠C = 52.13°
3. How can you reflect a figure in a coordinate plane?
Section 4.2
HSTX_GEOM_PE_04.02.indd 185
Reflections
185
6/19/14 6:03 PM
4.2 Lesson
What You Will Learn
Perform reflections.
Perform glide reflections.
Core Vocabul
Vocabulary
larry
Identify lines of symmetry.
reflection, p. 186
line of reflection, p. 186
glide reflection, p. 188
line symmetry, p. 189
line of symmetry, p. 189
Solve real-life problems involving reflections.
Performing Reflections
Core Concept
Reflections
A reflection is a transformation that uses a line like a mirror to reflect a figure.
The mirror line is called the line of reflection.
A reflection in a line m maps every point
P
P in the plane to a point P′, so that for
each point one of the following properties
is true.
• If P is not on m, then m is the
—, or
perpendicular bisector of PP′
m
P
P′
m
point P not on m
• If P is on m, then P = P′.
P′
point P on m
Reflecting in Horizontal and Vertical Lines
Graph △ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the
reflection described.
a. In the line n: x = 3
b. In the line m: y = 1
SOLUTION
a. Point A is 2 units left of line n, so its
reflection A′ is 2 units right of line n
at (5, 3). Also, B′ is 2 units left of
line n at (1, 2), and C′ is 1 unit right
of line n at (4, 1).
4
2
y
n
A
4
A′
B
B′
C
b. Point A is 2 units above line m, so
A′ is 2 units below line m at (1, −1).
Also, B′ is 1 unit below line m at
(5, 0). Because point C is on line m,
you know that C = C′.
y
A
2
C
C′
2
4
Monitoring Progress
C′
6
x
B
m
B′
6
A′
x
Help in English and Spanish at BigIdeasMath.com
Graph △ABC from Example 1 and its image after a reflection in the given line.
186
Chapter 4
HSTX_GEOM_PE_04.02.indd 186
1. x = 4
2. x = −3
3. y = 2
4. y = −1
Transformations
6/19/14 6:03 PM
Reflecting in the Line y = x
— with endpoints F(−1, 2) and G(1, 2) and its image after a reflection in the
Graph FG
line y = x.
REMEMBER
The product of the slopes
of perpendicular lines
is −1.
SOLUTION
The slope of y = x is 1. The segment from F to
—, is perpendicular to the line of
its image, FF′
— will be −1
reflection y = x, so the slope of FF′
(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).
y
4
F
y=x
G
G′
−2
4 x
F′
−2
— will also be −1. From G, move
The slope of GG′
0.5 unit right and 0.5 unit down to y = x. Then move
0.5 unit right and 0.5 unit down to locate G′(2, 1).
You can use coordinate rules to find the images of points reflected in four special lines.
Core Concept
Coordinate Rules for Reflections
• If (a, b) is reflected in the x-axis, then its image is the point (a, −b).
• If (a, b) is reflected in the y-axis, then its image is the point (−a, b).
• If (a, b) is reflected in the line y = x, then its image is the point (b, a).
• If (a, b) is reflected in the line y = −x, then its image is the point (−b, −a).
Reflecting in the Line y = −x
— from Example 2 and its image after a reflection in the line y = −x.
Graph FG
SOLUTION
y
Use the coordinate rule for reflecting in the line
y = −x to find the coordinates of the endpoints
— and its image.
of the image. Then graph FG
F
G
F′
(a, b) → (−b, −a)
2
F(−1, 2) → F′(−2, 1)
G′
G(1, 2) → G′(−2, −1)
Monitoring Progress
−2
x
y = −x
Help in English and Spanish at BigIdeasMath.com
The vertices of △JKL are J(1, 3), K(4, 4), and L(3, 1).
5. Graph △JKL and its image after a reflection in the x-axis.
6. Graph △JKL and its image after a reflection in the y-axis.
7. Graph △JKL and its image after a reflection in the line y = x.
8. Graph △JKL and its image after a reflection in the line y = −x.
—
9. In Example 3, verify that FF′ is perpendicular to y = −x.
Section 4.2
HSTX_GEOM_PE_04.02.indd 187
Reflections
187
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Performing Glide Reflections
Postulate
Postulate 4.2
Reflection Postulate
A reflection is a rigid motion.
m
E
D
F
E′
F′
Because a reflection is a rigid motion, and a rigid motion preserves length and angle
measure, the following statements are true for the reflection shown.
D′
• DE = D′E′, EF = E′F′, FD = F′D′
• m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′
Because a reflection is a rigid motion, the Composition Theorem (Theorem 4.1)
guarantees that any composition of reflections and translations is a rigid motion.
STUDY TIP
The line of reflection must
be parallel to the direction
of the translation to be a
glide reflection.
A glide reflection is a transformation involving a
translation followed by a reflection in which every
point P is mapped to a point P ″ by the following steps.
Q′
P′
Q″
P″
Step 1 First, a translation maps P to P′.
Step 2 Then, a reflection in a line k parallel to the
direction of the translation maps P′ to P ″.
Q
P
k
Performing a Glide Reflection
Graph △ABC with vertices A(3, 2), B(6, 3), and C(7, 1) and its image after the
glide reflection.
Translation: (x, y) → (x − 12, y)
Reflection: in the x-axis
SOLUTION
Begin by graphing △ABC. Then graph △A′B′C′ after a translation 12 units left.
Finally, graph △A″B″C″ after a reflection in the x-axis.
y
B′(−6, 3)
2
A′(−9, 2)
B(6, 3)
A(3, 2)
C′(−5, 1)
−12
−10
−8
−6
−4
C(7, 1)
−2
C″(−5, −1)
A″(−9, −2)
2
4
6
8
x
−2
B″(−6, −3)
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
10. WHAT IF? In Example 4, △ABC is translated 4 units down and then reflected in
the y-axis. Graph △ABC and its image after the glide reflection.
11. In Example 4, describe a glide reflection from △A″B″C ″ to △ABC.
188
Chapter 4
HSTX_GEOM_PE_04.02.indd 188
Transformations
6/19/14 6:03 PM
Identifying Lines of Symmetry
A figure in the plane has line symmetry when the figure can be mapped onto itself by
a reflection in a line. This line of reflection is a line of symmetry, such as line m at the
left. A figure can have more than one line of symmetry.
Identifying Lines of Symmetry
m
How many lines of symmetry does each hexagon have?
a.
b.
c.
b.
c.
SOLUTION
a.
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Determine the number of lines of symmetry for the figure.
12.
13.
14.
15. Draw a hexagon with no lines of symmetry.
Solving Real-Life Problems
Finding a Minimum Distance
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?
B
A
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.
Monitoring Progress
B
B′
m
A
C
m
Help in English and Spanish at BigIdeasMath.com
16. Look back at Example 6. Answer the question by using a reflection of point A
instead of point B.
Section 4.2
HSTX_GEOM_PE_04.02.indd 189
Reflections
189
6/19/14 6:03 PM
4.2
Exercises
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. VOCABULARY A glide reflection is a combination of which two transformations?
2. WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain
y
y
y
y
6
2
2
2
4
4
2
2
−4
x
−4
x
−2
x
−2
−2
−2
−2
x
2
Monitoring Progress and Modeling with Mathematics
In Exercises 3–6, determine whether the coordinate
plane shows a reflection in the x-axis, y-axis, or neither.
3.
11. J(2, 4), K(−4, −2), L(−1, 0); y = 1
4.
y
2
−4
A
B E
4
C
D
y
12. J(3, −5), K(4, −1), L(0, −3); y = −3
D
In Exercises 13–16, graph the polygon and its image
after a reflection in the given line. (See Examples 2
and 3.)
B
4x
A
−4
E
−2
−4
C
4x
13. y = x
F
−4
F
−6
5.
4
C
−4
−2
E
4
B
−2
A
D
A
2
4x
4
C
F x
−2
F
D
−4
B
C
y
2
B
4
2
4
2
14. y = x
y
6.
y
10. J(1, −1), K(3, 0), L(0, −4); x = 2
E
6
15. y = −x
A
y
4
2
2
7. J(2, −4), K(3, 7), L(6, −1); x-axis
8. J(5, 3), K(1, −2), L(−3, 4); y-axis
9. J(2, −1), K(4, −5), L(3, 1); x = −1
190
Chapter 4
HSTX_GEOM_PE_04.02.indd 190
D
A
x
16. y = −x
4
In Exercises 7–12, graph △JKL and its image after a
reflection in the given line. (See Example 1.)
D
4
−2
A
−4
C
x
−2
B
y
B
C
−4
2x
y
A
B
−2
4
−2
6x
C
−4
Transformations
6/19/14 6:03 PM
In Exercises 17–20, graph △RST with vertices R(4, 1),
S(7, 3), and T(6, 4) and its image after the glide
reflection. (See Example 4.)
17. Translation: (x, y) → (x, y − 1)
Reflection: in the y-axis
27. MODELING WITH MATHEMATICS You park at some
point K on line n. You deliver a pizza to House H,
go back to your car, and deliver a pizza to House J.
Assuming that you can cut across both lawns, how
can you determine the parking location K that
minimizes the distance HK + KJ ? (See Example 6.)
18. Translation: (x, y) → (x − 3, y)
Reflection: in the line y = −1
19. Translation: (x, y) → (x, y + 4)
J
H
Reflection: in the line x = 3
n
20. Translation: (x, y) → (x + 2, y + 2)
Reflection: in the line y = x
28. ATTENDING TO PRECISION Use the numbers and
In Exercises 21–24, determine the number of lines of
symmetry for the figure. (See Example 5.)
21.
symbols to create the glide reflection resulting in the
image shown.
22.
C″(−1, 5)
6
y
A″(5, 6)
4
B(−1, 1)
−4
23.
B″(4, 2)
2
A(3, 2)
−2
4
2
6
8x
−2
24.
−4
Translation: (x, y) →
Reflection: in y = x
C(2, −4)
(
)
,
25. USING STRUCTURE Identify the line symmetry
(if any) of each word.
a.
LOOK
b.
MOM
c.
OX
d.
1
x
26. ERROR ANALYSIS Describe and correct the error in
y
−8
A′
2
−6
B″
−4
2
−2
✗
B
−
In Exercises 29–32, find point C on the x-axis so
AC + BC is a minimum.
4
30. A(4, −5), B(12, 3)
6
A
8x
32. A(−1, 7), B(5, −4)
33. MATHEMATICAL CONNECTIONS The line y = 3x + 2
— to A″B″
— is a glide reflection.
AB
is reflected in the line y = −1. What is the equation of
the image?
Section 4.2
HSTX_GEOM_PE_04.02.indd 191
+
31. A(−8, 4), B(−1, 3)
B′
−2
y
3
29. A(1, 4), B(6, 1)
describing the transformation.
A″
2
Reflections
191
6/19/14 6:03 PM
34. HOW DO YOU SEE IT? Use Figure A.
35. CONSTRUCTION Follow these steps to construct a
reflection of △ABC in line m. Use a compass
and straightedge.
y
m
Step 1 Draw △ABC and line m.
Step 2 Use one compass setting
to find two points that are
equidistant from A on line
m. Use the same compass
setting to find a point on
the other side of m that is
the same distance from
these two points. Label
that point as A′.
x
Figure A
y
y
A
C
B
Step 3 Repeat Step 2 to find points B′ and C′.
Draw △A′B′C′.
36. USING TOOLS Use a reflective device to verify your
x
x
Figure 1
Figure 2
y
construction in Exercise 35.
37. MATHEMATICAL CONNECTIONS Reflect △MNQ in
the line y = −2x.
y
y = −2x
4
Q
−5
x
x
a. Which figure is a reflection of Figure A in the
line x = a? Explain.
b. Which figure is a reflection of Figure A in the
line y = b? Explain.
c. Which figure is a reflection of Figure A in the
line y = x? Explain.
d. Is there a figure that represents a glide reflection?
Maintaining Mathematical Proficiency
Use the diagram to find the angle measure.
41. m∠AOD
42. m∠BOE
43. m∠AOE
44. m∠COD
45. m∠EOD
46. m∠COE
47. m∠AOB
48. m∠COB
49. m∠BOD
Chapter 4
HSTX_GEOM_PE_04.02.indd 192
−3
38. THOUGHT PROVOKING Is the composition of a
translation and a reflection commutative? (In other
words, do you obtain the same image regardless of
the order in which you perform the transformations?)
39. MATHEMATICAL CONNECTIONS Point B′(1, 4) is the
image of B(3, 2) after a reflection in line c. Write an
equation for line c.
Reviewing what you learned in previous grades and lessons
(Section 1.5)
80 90 10 0
70 10 0 90 80 110 1
70 20
60 0 110
60 13
2
0
1
5 0
50 0
13
A
D
C
E
O
170 180
60
0 1 20 10 0
15
0 30
14 0
4
40. m∠AOC
192
1x
Figure 4
0 10
180 170 1 20 3
60
15 0 4
01 0
40
Figure 3
N
y
M
B
Transformations
6/19/14 6:03 PM
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