Download 5.7 Reflections and Symmetry

5.7
Goal
Reflections and Symmetry
A reflection is a transformation that creates a mirror image. The
original figure is reflected in a line that is called the line of reflection.
Identify and use
reflections and lines
of symmetry.
PROPERTIES OF REFLECTIONS
Key Words
1 The reflected image is congruent
●
• image p. 152
r
to the original figure.
• reflection
2 The orientation of the reflected
●
• line of symmetry
image is reversed.
3 The line of reflection is the
●
perpendicular bisector of
the segments joining the
corresponding points.
EXAMPLE
1
G
F
H
F
H
image
G
F
H
F
H
m
G
Solution
clockwise orientation
line of
reflection
Identify Reflections
Tell whether the red triangle
is the reflection of the blue
triangle in line m.
Visualize It!
original
Check to see if all three properties of a reflection are met.
1 Is the image congruent to the original figure? Yes. ✔
●
2 Is the orientation of the image reversed? Yes. ✔
●
TFGH has a clockwise orientation.
TFGH has a counterclockwise orientation.
3 Is m the perpendicular bisector of the segments connecting the
●
G
corresponding points? Yes. ✔
counterclockwise
orientation
To check, draw a
diagram and connect
the corresponding
endpoints.
G
F
H
F
H
m
G
ANSWER
282
Chapter 5
Congruent Triangles
Because all three properties are met, the red triangle is the
reflection of the blue triangle in line m.
EXAMPLE
Identify Reflections
2
m
Tell whether the red triangle is the
reflection of the blue triangle in line m.
Solution
Check to see if all three properties of a reflection are met.
1 Is the image congruent to the original figure? Yes. ✔
●
2 Is the orientation of the image reversed? No.
●
ANSWER
Student Help
EXAMPLE
Reflections in a Coordinate Plane
3
a. Which segment is the reflection
VOCABULARY TIP
Use the following
relationship to help
you remember that a
reflection is a flip:
The red triangle is not a reflection of the blue triangle.
&* in the x-axis? Which point
of AB
corresponds to A? to B?
A(4, 1)
J(4, 1)
b. Which segment is the reflection
&* in the y-axis? Which point
of AB
corresponds to A? to B?
reflection
flip
y E(1, 3)
B(1, 3)
1
D(4, 1)
1
x
K(1, 3)
Solution
&* and BK
&*, so the
a. The x-axis is the perpendicular bisector of AJ
&* in the x-axis is JK
&*.
reflection of AB
A is reflected onto J.
A(4, 1) → J(4, 1)
B(1, 3) → K(1, 3)
B is reflected onto K.
&* and BE
&*, so the
b. The y-axis is the perpendicular bisector of AD
&* in the y-axis is DE
&*.
reflection of AB
A is reflected onto D.
A(4, 1) → D(4, 1)
B(1, 3) → E(1, 3)
B is reflected onto E.
Identify Reflections
Tell whether the red figure is a reflection of the blue figure. If the red
figure is a reflection, name the line of reflection.
1.
2.
y
3.
y
y
1
1
1
1
1 x
1
x
x
5.7
Reflections and Symmetry
283
Symmetry In the photo, the mirror’s edge
creates a line of symmetry. A figure in the
plane has a line of symmetry if the figure
can be reflected onto itself by a reflection
in the line.
A line of symmetry is
a line of reflection.
Visualize It!
EXAMPLE
4
Determine Lines of Symmetry
Determine the number of lines of symmetry in a square.
Solution
You may want to draw
a shape on paper, cut
it out, and then fold it
to find the lines of
symmetry.
Think about how many different ways you can fold a square so that the
edges of the figure match up perfectly.
vertical fold
ANSWER
horizontal fold
diagonal fold
diagonal fold
A square has four lines of symmetry.
EXAMPLE
5
Determine Lines of Symmetry
Determine the number of lines of symmetry in each figure.
a.
b.
c.
Solution
a. 2 lines of symmetry
284
Chapter 5
Congruent Triangles
b. no lines of symmetry c. 6 lines of symmetry
EXAMPLE
Kaleidoscopes
6
Use Lines of Symmetry
Mirrors are used to create images seen through a kaleidoscope. The
angle between the mirrors is aA.
eyepiece
black glass
mirror
casing
mirror
mirror
glass
mirror
colored
glass or
liquid
cover
angle A
Top view
Image seen by viewer
KALEIDOSCOPES The parts
Find the angle measure used to create the kaleidoscope design. Use
of a kaleidoscope are shown
above.
the equation maA , where n is the number of lines of
Application Links
180
n
symmetry in the pattern.
CLASSZONE.COM
a.
b.
c.
Solution
a. The design has 3 lines of symmetry. So, in the formula, n 3.
180
n
180
3
ma A 60
b. The design has 4 lines of symmetry. So, in the formula, n 4.
180
n
180
4
maA 45
c. The design has 6 lines of symmetry. So, in the formula, n 6.
180
n
180
6
maA 30
Determine Lines of Symmetry
Determine the number of lines of symmetry in the figure.
4.
5.
6.
5.7
Reflections and Symmetry
285
5.7 Exercises
Guided Practice
Vocabulary Check
1. Complete the statement: A figure in the plane has a(n) __?__ if the
figure can be reflected onto itself by a(n) __?__ in the line.
Skill Check
Determine whether the red figure is a reflection of the blue figure.
2.
3.
4.
m
m
m
Flowers Determine the number of lines of symmetry in the flower.
5.
6.
7.
Practice and Applications
Extra Practice
Identifying Reflections Determine whether the figure in red is a
reflection of the figure in blue. Explain why or why not.
See p. 684.
8.
9.
m
10.
m
m
Reflections in a Coordinate Plane Tell whether the grid shows a
reflection in the x-axis, the y-axis, or neither.
Homework Help
11.
12.
y
D
B
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
Example 6:
286
Exs. 8–10
Exs. 8–10
Exs. 11–16
Exs. 21–29
Exs. 21–29
Exs. 37–39
Chapter 5
13.
y
C
D
D
3 x
A
Congruent Triangles
1
A
E
1 x
E
1
H
C
1
x
y
C
B
1
A
B
F
F
G
Student Help
SKILLS REVIEW
To review coordinates,
see p. 664.
Reflections in a Coordinate Plane In Exercises 14–16, use the
diagram at the right.
y
&* in the
14. Which segment is the reflection of AB
&* in the
15. Which segment is the reflection of AB
D
B
x-axis? Which point corresponds to A? to B?
A
y-axis? Which point corresponds to A ? to B?
&* with the
16. Compare the coordinates for AB
1
G
coordinates for its reflection in the x-axis.
How are the coordinates alike? How are
they different?
Visualize It!
C
1
x
E
H
F
Trace the figure and draw its reflection in line k.
17.
18.
19.
k
k
k
20. Paper Folding Follow these steps.
1 Fold a piece of paper in half, twice.
●
2 Draw a triangle and cut it out.
●
3 Unfold the paper and label the
●
sections.
B
C
A
D
Which of the triangles are reflections
of the triangle in section A? Explain.
Symmetry Decide whether the line shown is a line of symmetry.
21.
22.
23.
Lines of Symmetry Determine the number of lines of symmetry.
24.
25.
26.
5.7
Reflections and Symmetry
287
You be the Judge Determine whether all lines of symmetry are
shown. If not, sketch the figure and draw all the lines of symmetry.
27.
30.
28.
29.
Visualize It! A piece of paper is folded
in half and some cuts are made as shown.
Sketch the figure that represents the piece
of paper unfolded.
Careers
Type Design In Exercises 31 and 32, use the lowercase letters of the
alphabet shown below.
TYPE DESIGNERS design
fonts that appear in books,
magazines, newspapers, and
other materials that we read
every day. Jonathan Hoefler,
shown above, has designed
many fonts that are widely
used today.
Career Links
CLASSZONE.COM
31. Which letters are reflections of other letters?
32. Draw each letter that has at least one line of symmetry and sketch
its line(s) of symmetry. Which letters have one line of symmetry?
Which letters have two lines of symmetry?
Word Reflections Determine if the entire word has any lines of
symmetry. If so, write the word and draw the line(s) of symmetry.
33.
34.
35.
36.
Kaleidoscope Designs Find the measure of the angle between the
mirrors (aA) that produces the kaleidoscope design. Use the equation
180
n
maA .
37.
288
Chapter 5
Congruent Triangles
38.
39.
EXAMPLE
Show Triangles are Congruent
Show that TABC c TJKL.
y
B (5, 4)
A(1, 2)
C (5, 2)
Solution
Show that the corresponding sides
are congruent.
1
1
For sides on a horizontal grid line,
subtract the x-coordinates.
x
J (1, 2)
L(5, 2)
CA 5 1 4
LJ 5 1 4
K (5, 4)
For sides on a vertical grid line,
subtract the y-coordinates.
Student Help
BC 4 2 2
KL 4 (2) 2 2
For any other sides, use the distance formula.
LOOK BACK
AB (5
1
)2
(4
2
)2 4
2
22 20
For help with the
distance formula,
see p. 194.
JK (5
1
)2
((
4
(
2))
2 42
(2
)2 2
0
By the SSS Congruence Postulate, TABC c TJKL.
Showing Triangles are Congruent In Exercises 40 and 41, refer to
the example above. Show that TABC c TDEF.
40.
41.
y
y
C(3, 4)
B(2, 3)
1
A(2, 1)
1 D(2, 1)
1
C (6, 1)
F (6, 1) x
E(1, 1)
Standardized Test
Practice
1
x
D(1, 3)
F (4, 3)
E(2, 3)
B(1, 1) A(3, 1)
42. Multiple Choice Which triangle shows
y
the image when TXYZ is reflected in
the y-axis?
A
C
TDEF
TPQR
B
D
J
L
E
D
TJKL
F
Z
None of these
X
1
K
1
Y
Œ
R
x
P
43. Multiple Choice How many lines of symmetry
does the figure at the right have?
F
H
0
2
G
J
1
3
5.7
Reflections and Symmetry
289
Mixed Review
Showing Lines are Parallel Find the value of x so that p q.
(Lesson 3.5)
44.
45.
46.
p
105
82
q
p
(x 10)
x
p
q
92
(3x 1)
q
Finding Angle Measures Find the measure of a1. (Lesson 4.2)
47.
48.
1
38
Algebra Skills
49.
44
51
1
75
1
30
Comparing Numbers Compare the two numbers. Write the answer
using >, <, or . (Skills Review, p. 662)
50. 2348 and 2384
51. 5 and 7
52. 19.1 and 19.01
53. 11.2 and 11.238
54. 0.065 and 0.056
55. 1.011 and 1.11
Quiz 3
1. Sketch the overlapping triangles separately.
A
D
Mark all congruent angles and sides.
Which postulate or theorem can you use to
show that the triangles are congruent?
(Lesson 5.5)
B
C
Use the diagram to find the indicated measure(s). (Lesson 5.6)
2. Find DC.
3. Find ML and JK.
4. Find AB.
B
L
A
4
B
M
2
9
D
4
3x 4
25
J
C
A
2x 7
D
K
Determine the number of lines of symmetry in the figure. (Lesson 5.7)
5.
290
Chapter 5
Congruent Triangles
6.
7.
C