example 1 use similarity statements b. check that the ratios of corresponding side lengths are...
DESCRIPTION
GUIDED PRACTICE for Example 1 SOLUTION J = ~ ~~ == P,P, K Q LR and The congruent angles are JK PQ = KL QR = LJ RP The ratios of the corresponding side lengths are. L J K R P Q 1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality.TRANSCRIPT
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.
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 = 53b.
; ST = 30
18 = 53YZ ;
GUIDED PRACTICE for Example 1
SOLUTION
J =~ ~~ ==P, K Q L RandThe congruent angles are
JKPQ = KL
QR = LJRP
The ratios of the corresponding side lengths are.
L
J K
R
P Q
1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding
side lengths in a statement of proportionality.
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.
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 angels, so W J. So, the corresponding angles are congruent.
EXAMPLE 2 Find the scale factor
SOLUTION
STEP 2
Show that corresponding side lengths are proportional.
XWHJ
ZYFG
YXGH
WZJF
2520=
1512=
54=
2016== 5
4
54=
= 54
3024=
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 toFGHJ is
ANSWER
54
.
EXAMPLE 3 Use similar polygons
In the diagram, ∆DEF ~ ∆MNP. Find the value of x.
ALGEBRA
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
GUIDED PRACTICE for Examples 2 and 3
In the diagram, ABCD ~ QRST.
SOLUTION
STEP 1
Identify pairs of congruent angles. From the diagram, you can see that A = Q , T = D, and B = R.Angles C and S are right angles. So, all the corresponding angles are congruent.
2. What is the scale factor of QRST to ABCD ?
GUIDED PRACTICE for Examples 2 and 3
STEP 2
Show that corresponding side lengths are proportional.
QRAB
QTAD
TSDC
RSBC
510=
612=
12= 8
16=
4 x== 1
2
12=
GUIDED PRACTICE for Examples 2 and 3
The ratios are equal, so the corresponding side lengths are proportional.
ANSWER
So QRST ~ ABCD. The scale factor of QRST to ABCD 12
.
GUIDED PRACTICE for Examples 2 and 3
3. Find the value of x.
In the diagram, ABCD ~ QRST.
Write proportion.
Substitute.
Cross Products Property
Solve for x.
SOLUTIONThe triangles are similar, so the corresponding side lengths are proportional.
RSQS
BCAC=
4 6 4+ = x
12 x+
4(12 + x) = 10 xx = 8
GUIDED PRACTICE for Examples 2 and 3
ANSWER
So the value of x is 8
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.
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.
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.
GUIDED PRACTICE for Example 4
4. Find the scale factor of FGHJK to ABCDE.
In the diagram, ABCDE ~ FGHJK.
The scale factor is the ratio of the
length is
ANSWER
1510 = 3
2
GUIDED PRACTICE for Example 4
5. Find the value of x.
In the diagram, ABCDE ~ FGHJK.
x 18
1015= Use Theorem 6.1 to write a
proportion.
Cross product property.
You can use the theorem 6.1 to find the perimeter of x
x = 12
SOLUTION
15 x = 18 10
GUIDED PRACTICE for Example 4
ANSWER
The value of x is 12
GUIDED PRACTICE for Example 4
6. Find the perimeter of ABCDE.
In the diagram, ABCDE ~ FGHJK.
SOLUTION
As the two polygons are similar the corresponding side lengths are similar To find the perimeter of ABCDE first find its’ side lengths.
GUIDED PRACTICE for Example 4
FG AB = FK
AEWrite Equation
Substitute
15x = 180 Cross Products Property
x = 12 Solve for x
AE = 12
To find AE
1510 = 18
x
GUIDED PRACTICE for Example 4
Write Equation
1510 = 15
ySubstitute
15y = 150 Cross Products Property
y = 10 Solve for y
ED = 10
To find ED
FG AB = KJ
ED
GUIDED PRACTICE for Example 4
FG AB = HJ
CDWrite Equation
1510 = 12
zSubstitute
15z = 120 Cross Products Property
z = 8 Solve for z
DC = 8
To find DC
GUIDED PRACTICE for Example 4
FG AB = GH
BCWrite Equation
1510 = 9
aSubstitute
15a = 90 Cross Products Property
a = 6 Solve for x
BC = 6
To find BC
GUIDED PRACTICE for Example 4
The perimeter of ABCDE = AB + BC + CD + DE + EA= 10 + 6 + 8 + 10 + 12= 46
ANSWER The perimeter of ABCDE = 46
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
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.
The length of the altitude PS is 15.
Write proportion.
Substitute 20 for PY.
Multiply each side by 20 and simplify.
PSPY
3 4=
PS20
3 4=
=PS 15
ANSWER
GUIDED PRACTICE for Example 5
In the diagram, ABCDE ~ FGHJK.
In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM.
7.
GUIDED PRACTICE for Example 5
JLEG
First find the scale factor of ∆ JKL to ∆ EFG.
SOLUTION
48 + 48= 40 + 40 = 9680 = 6
5
Because the ratio of the lengths of the median in similar triangles is equal to the scale factor, you can write the following proportion.
GUIDED PRACTICE for Example 5
SOLUTION
KM = 42
KMHF = 6
5
KM35
= 65
Write proportion.
Substitute 35 for HF.
Multiply each side by 35 and simplify.