1.
6.3 Use Similar Polygons
6.3
Bell Thinger
1. Solve 12 = x
10
60
ANSWER 72
2. The scale of a map is 1 cm : 10 mi. The actual
distance between two towns is 4.3 miles. Find the
length on the map.
ANSWER 0.43 cm
3. A model train engine is 9 centimeters long. The
actual engine is 18 meters long. What is the scale of
the model?
ANSWER
1 cm : 2 m
3.
Example 1
6.3
In the diagram, ∆RST ~ ∆XYZ
a. List all pairs of
congruent angles.
b. Check that the ratios of
corresponding side
lengths are equal.
c. Write the ratios of the corresponding side
lengths in a statement of proportionality.
SOLUTION
a. R ≅ X, S ≅ Y and T ≅ Z
4.
Example 1
6.3
In the diagram, ∆RST ~ ∆XYZ
a. List all pairs of
congruent angles.
b. Check that the ratios of
corresponding side
lengths are equal.
c. Write the ratios of the corresponding side
lengths in a statement of proportionality.
SOLUTION
b.
RS = 20
5 ; ST = 30 = 5 ; TR = 25 = 5
3
15
3
ZX
18
XY
12 = 3
YZ
5.
Example 1
6.3
In the diagram, ∆RST ~ ∆XYZ
a. List all pairs of
congruent angles.
b. Check that the ratios of
corresponding side
lengths are equal.
c. Write the ratios of the corresponding side
lengths in a statement of proportionality.
SOLUTION
c. Because the ratios in part (b) are equal,
RS = ST = TR .
YZ
ZX
XY
6.
Guided Practice
6.3
1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent
angles. Write the ratios of the corresponding side lengths
in a statement of proportionality.
ANSWER
∠J ≅ ∠P, ∠K ≅ ∠Q and ∠L ≅ ∠R ;
JK = KL
LJ
= RP
PQ
QR
7.
Example 2
6.3
Determine whether the polygons are similar. If they
are, write a similarity statement and find the scale
factor of ZYXW to FGHJ.
8.
Example 2
6.3
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.
9.
Example 2
6.3
SOLUTION
STEP 2
Show that corresponding side lengths are proportional.
ZY
=
FG
25
20
XW
=
HJ
15
12
=
5
4
YX =
GH
30
24
=
5
4
WZ =
JF
5
20
=
4
16
= 5
4
10.
Example 2
6.3
SOLUTION
The ratios are equal, so the corresponding side lengths
are proportional.
So ZYXW ~ FGHJ. The scale factor of ZYXW to
FGHJ is 5 .
4
11.
Example 3
6.3
ALGEBRA In the diagram,
∆DEF ~ ∆MNP. Find the value
of x.
SOLUTION
The triangles are similar, so the corresponding side
lengths are proportional.
MN = NP
EF
DE
12 = 20
x
9
12x = 180
x = 15
Write proportion.
Substitute.
Cross Products Property
Solve for x.
12.
Guided Practice
6.3
In the diagram, ABCD ~ QRST.
2.
What is the scale factor of QRST to ABCD ?
ANSWER
3.
1
2
Find the value of x.
ANSWER
8
14.
Example 4
6.3
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.
a. Find the scale factor of the new pool to an
Olympic pool.
b. Find the perimeter of an Olympic pool and the
new pool.
15.
Example 4
6.3
SOLUTION
a.
Because the new pool will be similar to an Olympic
pool, the scale factor is the ratio of the lengths,
4.
40
=
5
50
b. 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.
x
= 4
Use Theorem 6.1 to write a proportion.
150
5
x = 120
Multiply each side by 150 and simplify.
The perimeter of the new pool is 120 meters.
16.
Guided Practice
6.3
In the diagram, ABCDE ~ FGHJK.
4. Find the scale factor of
FGHJK to ABCDE.
ANSWER
3
2
Find the value of x.
ANSWER
12
ANSWER
48
5.
6. Find The perimeter of ABCDE.
18.
Example 5
6.3
In the diagram, ∆TPR ~ ∆XPZ. Find the length of the
altitude PS .
SOLUTION
First, find the scale factor of ∆TPR to ∆XPZ.
TR = 6 + 6 = 12 = 3
8+8
4
16
XZ
19.
Example 5
6.3
Because the ratio of the lengths of the altitudes in
similar triangles is equal to the scale factor, you can
write the following proportion.
=
3
4
Write proportion.
PS
20
=
3
4
Substitute 20 for PY.
PS
= 15
PS
PY
Multiply each side by 20 and simplify.
The length of the altitude PS is 15.
20.
Guided Practice
6.3
7. In the diagram, ∆JKL ~ ∆ EFG. Find the length of
the median KM.
ANSWER
42
21.
Exit Slip
6.3
1.
Determine whether the polygons are similar. If
they are, write a similarity statement and find the
scale factor of EFGH to KLMN.
ANSWER
Yes; EFGH ~ KLMN; the scale factor is
2:1
22.
Exit Slip
6.3
2. In the diagram,
of x.
ANSWER
13.5
DEF ~
HJK. Find the value
23.
Exit Slip
6.3
3.
Two similar triangles have the scale factor 5 : 4.
Find the ratio of their corresponding altitudes and
median.
ANSWER
4.
5:4;5:4
Two similar triangles have the scale factor 3 : 7.
Find the ratio of their corresponding perimeters
and areas.
ANSWER
3 : 7; 9: 49
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