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Transcript
Similar Figures
Octahedral fluorite is a crystal found in nature. It
grows in the shape of an octahedron, which is a
solid figure with eight triangular faces. The
triangles in different-sized fluorite crystals are
similar figures. Similar figures have the same
shape but not necessarily the same size.
5-5 C2: Similar Figures and
Proportions
Matching sides of two or more polygons are
called corresponding sides, and matching
angles are called corresponding angles.
Corresponding
sides
B
A
C
D
E
F
Corresponding angles
SIMILAR FIGURES
Tickmarks are symbols used to help
you determine:
• Congruence of sides and angles
• Right angles
• Parallel lines
If two figures are similar, then the measures of
the corresponding angles are equal and the ratios
of the lengths of the corresponding sides are
proportional.
To find out if triangles are similar, determine
whether the ratios of the lengths of their
corresponding sides are proportional. If the ratios
are proportional, then the corresponding angles
must have equal measures.
1
Similar Figures
In order for figures to be similar 2 things must
be true:
1. Corresponding angles must be congruent
2. Corresponding sides must be proportional
Reading Math
A side of a figure can be named by its endpoints,
with a bar above.
AB
Without the bar, the letters indicate the length of
the side.
When matching corresponding sides be
sure to list segments in the correct order.
Additional Example 1: Determining Whether Two
Triangles Are Similar
Try This: Example 1
Identify the corresponding sides in the pair of triangles.
Then use ratios to determine whether the triangles are
similar.
E
AB corresponds to DE.
16 in
A 10 in C
28 in
BC corresponds to EF.
D
4 in
7 in
40 in
F
AC corresponds to DF.
B
Identify the corresponding sides in the pair of triangles.
Then use ratios to determine whether the triangles are
similar.
E
AB corresponds to DE.
9 in
9 in C
A
21 in
BC corresponds to EF.
D
3 in
7 in
27 in
F
AC corresponds to DF.
B
AB ? BC ? AC
=
=
Write ratios using the corresponding sides.
DE
EF
DF
4 ? 7 ? 10
Substitute the length of the sides.
=
=
16
28 40
? 1 ? 1
1 =
Simplify each ratio.
=
4
4
4
Since the ratios of the corresponding sides are equivalent, the
triangles are similar.
AB ? BC ? AC
=
=
Write ratios using the corresponding sides.
DE
EF
DF
3 ? 7 ? 9
Substitute the length of the sides.
=
=
9
21 27
? 1 ? 1
1 =
Simplify each ratio.
=
3
3
3
Since the ratios of the corresponding sides are equivalent, the
triangles are similar.
In figures with four or more sides, it is possible
for the corresponding side lengths to be
proportional and the figures to have different
shapes. To find out if these figures are similar,
first check that their corresponding angles have
equal measures.
8m
10 m
4m
4m
5m
Additional Example 2: Determining Whether Two
Four-Sided Figures are Similar
Use the properties of similarity to determine
whether the figures are similar.
The corresponding angles of the
figures have equal measure.
5m
8m
10 m
10 m = 5 m
8m 4m
Write each set of
corresponding sides as a ratio.
2
Additional Example 2 Continued
Additional Example 2 Continued
Determine whether the ratios of the lengths of the
corresponding sides are proportional.
MN
MN corresponds to QR.
QR
? corresponding
? MP
? NO
OP =
MNratios
Write
using
=
=
ST
QR
RS
QT
sides.
NO NO corresponds to RS.
RS
OP
ST
6 ? 8 ? 4 ? 10
Substitute
the length of the
9 = 12 = 6 = 15
sides.
OP corresponds to ST.
2? 2? 2? 2
= = ratio.
=
Simplify
3 each
3
3 3
MP MP corresponds to QT.
QT
Since the ratios of the corresponding sides
are equivalent, the figures are similar.
Try This: Example 2
Try This: Example 2 Continued
Use the properties of similarity to determine
whether the figures are similar.
M
P
100 m
80°
60 m
M
65°
47.5 m
90°
125°
80 m
N
Q
O
T
65°
240 m
190 m
90°
O
T
65°
240 m
R
MN
MN corresponds to QR.
QR
NO NO corresponds to RS.
RS
400 m
OP
ST
OP corresponds to ST.
190 m
125°
90°
S
320 m
125°
80 m
80°
125°
90°
65°
47.5 m
Q
Write each set of
corresponding sides as a ratio.
P
100 m
80°
60 m
N
400 m
80°
R
The corresponding angles of the
figures have equal measure.
320 m
S
MP MP corresponds to QT.
QT
Try This: Example 2 Continued
Determine whether the ratios of the lengths of the
corresponding sides are proportional.
M
P
100 m
80°
60 m
65°
47.5 m
90°
125°
80 m
N
Q
T
65°
240 m
190 m
125°
90°
R
320 m
8-4 C1: Similar Figures
O
400 m
80°
? corresponding
? MP
? NO
OP =
MNratios
Write
using
=
=
sides.
ST
QR
RS
QT
60 ? 80
? 47.5of
Substitute
the length
the
? 100
240 = 320 = 190 = 400
sides.
1? 1? 1? 1
4 4
4 4
= =each=ratio.
Simplify
S Since the ratios of the corresponding sides
are equivalent, the figures are similar.
3
Insert Lesson Title Here
Similar Figures
Vocabulary
similar
corresponding sides
corresponding angles
Learn to use ratios to identify similar
figures.
Course 1
Two or more figures are similar if they have
exactly the same shape. Similar figures may
be different sizes.
A
2 cm
B
Similar figures have corresponding sides and
corresponding angles.
• Corresponding sides have
are proportional.
lengths that
• Corresponding angles are congruent.
3 cm
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
X
Corresponding sides:
AB corresponds to WX.
BC corresponds to XY.
CD corresponds to YZ.
AD corresponds to WZ.
9 cm
Y
Corresponding angles:
A corresponds to
B corresponds to
W.
X.
C corresponds to
D corresponds to
Y.
Z.
Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Additional Example 1: Finding Missing
Measures in Similar Figures
Z
6 cm
The two triangles are similar. Find the
missing length y and the measure of D.
C
X
9 cm
Y
In the rectangles above, one proportion is
AB
AD
2
3
=
, or
= .
WX WZ
6
9
If you cannot use corresponding side lengths to
write a proportion, or if corresponding angles are
not congruent, then the figures are not similar.
100
111
Write a proportion using
____
= ___
200
y
corresponding side lengths.
200 • 111 = 100 • y The cross products are equal.
Course 1
4
Additional Example 1 Continued
Try This: Example 1
The two triangles are similar. Find the
missing length y and the measure of B.
The two triangles are similar. Find the
missing length y and the measure of D.
22,200 = 100y
22,200 = ____
100y
______
100
100
A
60 m
65°
50 m
45° 52 m
y is multiplied by 100.
Divide both sides by 100
to undo the multiplication.
B
120 m
100 m
y
222 mm = y
Angle D is congruent to angle C, and m
m
D = 70°
100 • 52 = 50 • y
Try This: Example 1 Continued
5,200 = 50y
_____
___
50
50
104 m = y
Additional Example 2: Problem Solving Application
Divide both sides by 50 to
undo the multiplication.
The answer will be the width of the actual painting.
1
Understand the Problem
List the important information:
A=
B = 65°
• The actual painting and the reduction above are similar.
• The reduced painting is 2 cm tall and 3 cm wide.
• The actual painting is 54 cm tall.
Additional Example 2 Continued
2
Make a Plan
Additional Example 2 Continued
3
Draw a diagram to represent the situation.
Use the corresponding sides to write a
proportion.
Actual
Reduced
2
The cross products are equal.
y is multiplied by 50.
Angle B is congruent to angle A, and m
65°.
m
Write a proportion using
corresponding side lengths.
This reduction is similar to a
picture that Katie painted. The
height of the actual painting is
54 centimeters. What is the
width of the actual painting?
The two triangles are similar. Find the
missing length y and the measure of B.
5,200 = 50y
50
52
____
= ___
100
y
C = 70°.
Solve
2 cm
3 cm
_____
=
w cm Write a proportion.
54 cm
54 • 3 = 2 • w The cross products are equal.
162 = 2w
162
2w
____
= ___
2
2
81 = w
54
3
w
w is multiplied by 2.
Divide both sides by 2 to
undo the multiplication.
The width of the actual painting is 81 cm.
5
Try This: Example 2
Additional Example 2 Continued
4
Look Back
Estimate to check your answer. The ratio of
the heights is about 2:50 or 1:25. The ratio
of the widths is about 3:90, or 1:30. Since
these ratios are close to each other, 81 cm
is a reasonable answer.
This reduction is similar to a
picture that Marty painted. The
height of the actual painting is 3 in.
39 inches. What is the width of
the actual painting?
1
4 in.
Understand the Problem
The answer will be the width of the actual painting.
List the important information:
• The actual painting and the reduction above are similar.
• The reduced painting is 3 in. tall and 4 in. wide.
• The actual painting is 39 in. tall.
Try This: Example 2 Continued
2
Make a Plan
Try This: Example 2 Continued
3
Draw a diagram to represent the situation.
Use the corresponding sides to write a
proportion.
Solve
3 in
4 in
_____
= ____ Write a proportion.
39 in
w in
39 • 4 = 3 • w The cross products are equal.
Actual
Reduced
3
156 = 3w
3w
156 = ___
____
3
3
52 = w
39
4
w
w is multiplied by 3.
Divide both sides by 3 to
undo the multiplication.
The width of the actual painting is 52 inches.
Try This: Example 2 Continued
4
Look Back
Estimate to check your answer. The ratio of
the heights is about 4:40, or 1:10. The
ratio of the widths is about 5:50, or 1:10.
Since these ratios are the same, 52 inches
is a reasonable answer.
6