6.3 Similar Polygons
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6.3 Similar Polygons
Objective: Use proportions to identify
similar polygons
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EXAMPLE 1 Use similarity statements
b. Check that the ratios of corresponding side lengths are equal.
In the diagram, ∆RST ~ ∆XYZ
a. List all pairs of congruentangles.
c. Write the ratios of the corresponding sidelengths in a statement of proportionality.
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EXAMPLE 1 Use similarity statements
TRZX = 25
15 = 53
c. Because the ratios in part (b) are equal,
SOLUTION
YZRSXY = ST = TR
ZX.
a. R =~ ~~ ==X, S Y T Zand
RSXY = 20
12 = 53
b.
; ST = 3018 = 5
3YZ;
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GUIDED PRACTICE for Example 1
1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding
side lengths in a statement of proportionality.
J =~ ~~ ==P, K Q L Rand JKPQ = KL
QR = LJRP;
ANSWER
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EXAMPLE 2 Find the scale factor
Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ZYXW to FGHJ.
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EXAMPLE 2 Find the scale factor
SOLUTION
STEP 1
Identify pairs of congruent angles. From the diagram, you can see that Z F, Y G, and X H.Angles W and J are right angles, so W J. So, the corresponding angles are congruent.
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EXAMPLE 2 Find the scale factor
SOLUTION
STEP 2
Show that corresponding side lengths are proportional.
XWHJ
ZYFG
YXGH
WZJF
2520
=
1512
=
54
=
2016
== 54
54
=
=54
3024
=
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EXAMPLE 2 Find the scale factor
SOLUTION
The ratios are equal, so the corresponding side lengths are proportional.
So ZYXW ~ FGHJ. The scale factor of ZYXW to
FGHJ is
ANSWER
54
.
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EXAMPLE 3 Use similar polygons
In the diagram, ∆DEF ~ ∆MNP. Find the value of x.
ALGEBRA
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EXAMPLE 3 Use similar polygons
Write proportion.
Substitute.
Cross Products Property
Solve for x.
SOLUTION
The triangles are similar, so the corresponding side lengths are proportional.
x = 15
12x = 180
MNDE
NPEF=
=129
20x
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GUIDED PRACTICE for Examples 2 and 3
In the diagram, ABCD ~ QRST.
2. What is the scale factor of QRST to ABCD ?
1 2
ANSWER
3. Find the value of x.
ANSWER 8
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EXAMPLE 4 Find perimeters of similar figures
Swimming
A town is building a new swimming pool. An Olympic pool is rectangular with length 50 meters and width 25 meters. The new pool will be similar in shape, but only 40 meters long.
Find the scale factor of the new pool to an Olympic pool.
a.
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EXAMPLE 4 Find perimeters of similar figures
SOLUTION
Because the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths,
a. 40
50=
45
Find the perimeter of an Olympic pool and the new pool.
b.
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EXAMPLE 4 Find perimeters of similar figures
x150
45
= Use Theorem 6.1 to write a proportion.
x = 120 Multiply each side by 150 and simplify.
The perimeter of the new pool is 120 meters.
ANSWER
The perimeter of an Olympic pool is 2(50) + 2(25) = 150 meters. You can use Theorem 6.1 to find the perimeter x of the new pool.
b.
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GUIDED PRACTICE for Example 4
4. Find the scale factor of FGHJK to ABCDE.
In the diagram, ABCDE ~ FGHJK.
3 2
ANSWER
5. Find the value of x. ANSWER 12
6. Find The perimeter of ABCDE. ANSWER 46
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EXAMPLE 5 Use a scale factor
In the diagram, ∆TPR ~ ∆XPZ. Find the length of the altitude PS .
SOLUTION
First, find the scale factor of ∆TPR to ∆XPZ.
TRXZ
6 + 6= 8 + 8 = 1216
= 3 4
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EXAMPLE 5 Use a scale factor
Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion.
Write proportion.
Substitute 20 for PY.
Multiply each side by 20 and simplify.
PSPY
3 4=
PS20
3 4
=
=PS 15
The length of the altitude PS is 15.
ANSWER
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GUIDED PRACTICE for Example 5
In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM.
7.
ANSWER 42