Similar polygons

1
Similarity
2
Ratios

• Ratio a comparison of two quantities
• can be expressed as

3
Ratios

• Extended Ratiosused to compare 3 or more
  numbers. abc means
• the ratio of the first two number is ab
• the ratio of the last two numbers is bc
• the ratio of first and last numbers is ac

4

• The lengths of the sides of a triangle are the
  extended ratio 356. The perimeter of the
  triangle is 98 in. What is the length of the
  longest side?

35
21
42
5
Similar Polygons
Definition Two polygons are similar if 1.
Corresponding angles are congruent. 2.
Corresponding sides are in proportion.
Two polygons are similar if they have the same
shape not necessarily have the same size.
The scale factor is the ratio between a pair of
corresponding sides.
Scale Factor
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Naming Similar Polygons
When naming similar polygons, the vertices
(angles, sides) must be named in the
corresponding order.
P
Q
A
B
C
D
S
R
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Example-
The two polygons are similar. Solve for x, y and
z.
Step1 Write the proportion of the sides.
Step 2 Replace the proportion with values.
Step 3 Find the scale factor between the two
polygons.
Note The scale factor has the larger
quadrilateral in the numerator and the smaller
quadrilateral in the denominator.
Step 4 Write separate proportions for each
missing side and solve.
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Example
If ?ABC ?ZYX, find the scale factor from ?ABC
to ?ZYX.
Scale factor is same as the ratio of the sides.
Always put the first polygon mentioned in the
numerator.
The scale factor from ?ABC to ?ZYX is 2/1.
½
What is the scale factor from ?ZYX to ?ABC?
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EXAMPLE 1
Use similarity statements
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EXAMPLE 1
Use similarity statements
SOLUTION
c. Because the ratios in part (b) are equal,
11
for Example 1
GUIDED PRACTICE
SOLUTION
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EXAMPLE 2
Find the scale factor
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EXAMPLE 2
Find the scale factor
SOLUTION
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EXAMPLE 2
Find the scale factor
SOLUTION
STEP 2
Show that corresponding side lengths are
proportional.
15
EXAMPLE 2
Find the scale factor
SOLUTION
The ratios are equal, so the corresponding side
lengths are proportional.
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EXAMPLE 3
Use similar polygons
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EXAMPLE 3
Use similar polygons
SOLUTION
The triangles are similar, so the corresponding
side lengths are proportional.
Write proportion.
Substitute.
12x 180
Cross Products Property
x 15
Solve for x.
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for Examples 2 and 3
GUIDED PRACTICE
In the diagram, ABCD QRST.
2. What is the scale factor of QRST to ABCD ?
STEP 1
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for Examples 2 and 3
GUIDED PRACTICE
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for Examples 2 and 3
GUIDED PRACTICE
The ratios are equal, so the corresponding side
lengths are proportional.
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for Examples 2 and 3
GUIDED PRACTICE
3. Find the value of x.
SOLUTION
The triangles are similar, so the corresponding
side lengths are proportional.
Write proportion.
Substitute.
Cross Products Property
Solve for x.
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for Examples 2 and 3
GUIDED PRACTICE
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EXAMPLE 4
Find perimeters of similar figures
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EXAMPLE 4
Find perimeters of similar figures
SOLUTION
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EXAMPLE 4
Find perimeters of similar figures
Use Theorem 6.1 to write a proportion.
x 120
Multiply each side by 150 and simplify.
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for Example 4
GUIDED PRACTICE
4. Find the scale factor of FGHJK to ABCDE.
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for Example 4
GUIDED PRACTICE
5. Find the value of x.
SOLUTION
You can use the theorem 6.1 to find the perimeter
of x
Use Theorem 6.1 to write a proportion.
Cross product property.
x 12
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for Example 4
GUIDED PRACTICE
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for Example 4
GUIDED PRACTICE
6. Find the perimeter of ABCDE.
SOLUTION
As the two polygons are similar the corresponding
side lengths are similar
To find the perimeter
of ABCDE first find its side lengths.
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for Example 4
GUIDED PRACTICE
To find AE
Write Equation
Substitute
15x 180
Cross Products Property
x 12
Solve for x
AE 12
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for Example 4
GUIDED PRACTICE
To find ED
Write Equation
Substitute
15y 150
Cross Products Property
y 10
Solve for y
ED 10
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for Example 4
GUIDED PRACTICE
To find DC
Write Equation
Substitute
15z 120
Cross Products Property
z 8
Solve for z
DC 8
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for Example 4
GUIDED PRACTICE
To find BC
Write Equation
Substitute
15a 90
Cross Products Property
a 6
Solve for x
BC 6
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for Example 4
GUIDED PRACTICE
The perimeter of ABCDE AB BC CD DE EA
10 6 8 10 12
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