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10-1
Ratio Proportion and Similarity
Homework:
Chapter 10-1
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GEOMETRY
10-1
Ratio Proportion and Similarity
Warm Up
Find the slope of the line through each pair of
points.
1. (1, 5) and (3, 9)
2
2. (–6, 4) and (6, –2)
Solve each equation.
3. 4x + 5x + 6x = 45 x = 3
4. (x – 5)2 = 81
5. Write
x = 14 or x = –4
in simplest form.
GEOMETRY
10-1
Ratio Proportion and Similarity
Objectives
Write and simplify ratios.
Use proportions to solve problems.
GEOMETRY
10-1
Ratio Proportion and Similarity
The Lord of the Rings movies transport viewers
to the fantasy world of Middle Earth. Many
scenes feature vast fortresses, sprawling cities,
and bottomless mines. To film these images,
the moviemakers used ratios to help them
build highly detailed miniature models.
GEOMETRY
10-1
Ratio Proportion and Similarity
A ratio compares two numbers by division. The
ratio
a:b
, of two numbers a and b can be
written as a to b, or
a
b
, where b ≠ 0. For
example, the ratios 1 to 2, 1:2, and
all
represent the same comparison.
GEOMETRY
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Ratio Proportion and Similarity
Remember!
In a ratio, the denominator of the fraction cannot be
zero because division by zero is undefined.
GEOMETRY
10-1
Ratio Proportion and Similarity
Ratios & Slope
Write a ratio expressing the slope of l.
Substitute the
given values.
Simplify.
GEOMETRY
10-1
Ratio Proportion and Similarity
TEACH! Ratio & Slope
Given that two points on m are C(–2, 3) and D(6, 5),
write a ratio expressing the slope of m.
Substitute the
given values.
Simplify.
GEOMETRY
10-1
Ratio Proportion and Similarity
Example 1 : Similarity statement & Ratios
Are the triangles similar? If they are, write a similarity
statement and give the ratio. If they are not, explain why.
E
The corresponding angle
pairs are congruent.
Find the ratios and compare.
AC 18 3


DF 24 4
ABC
AB 15 3


FE 20 4
B
15
A
18
C
16
20
12
F
D
24
BC 12 3


ED 16 4
3
FED with a similarity ratio , or 3:4.
4
GEOMETRY
10-1
Ratio Proportion and Similarity
A Golden Rectangle is a rectangle that can be divided
into a square and a rectangle that is similar to the
original rectangle. The ratio of the length of a Golden
Rectangle to its width is called the Golden Ratio.
C
F
D
E
F
B
A
ABCD
BCFE
E
C
B
Definition of Golden Rectangle.
GEOMETRY
10-1
Ratio Proportion and Similarity
A ratio can involve more than two numbers. For the
rectangle, the ratio of the side lengths may be
written as 3:7:3:7.
GEOMETRY
10-1
Ratio Proportion and Similarity
Example 2: Using Ratios
The ratio of the side lengths of a triangle is 4:7:5,
and its perimeter is 96 cm. What is the length of
the shortest side?
Let the side lengths be 4x, 7x, and 5x.
Then 4x + 7x + 5x = 96 .
After like terms are combined,
16x = 96.
So x = 6.
The length of the shortest side is
4x = 4(6) = 24 cm.
GEOMETRY
10-1
Ratio Proportion and Similarity
TEACH! Example 2
The ratio of the angle measures in a
triangle is 1:6:13. What is the measure of
each angle? A  B  C  180o
1x + 6x + 13x = 180°
20x = 180°
x = 9°
B = 6x
B = 6(9°)
C = 13x
C = 13(9°)
B = 54°
C = 117°
GEOMETRY
10-1
Ratio Proportion and Similarity
-A proportion is an equation stating that
two ratios are equal. In the proportion,
the values
a and d are the
extremes.
-The values b and c are the means.
- An extended proportion is when three
or more ratios are equal, such as:
8 15 12


24 20 16
GEOMETRY
10-1
Ratio Proportion and Similarity
In Algebra 1 you learned the Cross Products Property.
The product of the extremes ad and the product of the
means bc are called the cross products.
GEOMETRY
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Ratio Proportion and Similarity
Reading Math
The Cross Products Property can also
be stated as,
“In a proportion, the product of the
extremes is equal to the product of
the means.”
GEOMETRY
10-1
Ratio Proportion and Similarity
Example 3: Solving Proportions
Solve the proportion.
(z – 4)2 = 5(20)
Cross Products Property
(z – 4)2 = 100
Simplify.
(z – 4) = 10
Find the square root of both sides.
(z – 4) = 10 or (z – 4) = –10
Rewrite as two eqns.
z = 14 or z = –6
Add 4 to both sides.
GEOMETRY
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Ratio Proportion and Similarity
TEACH! Example 3A
Solve the proportion.
3(56) = 8(x)
168 = 8x
x = 21
Cross Products Property
Simplify.
Divide both sides by 8.
GEOMETRY
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Ratio Proportion and Similarity
TEACH! Example 3B
Solve the proportion.
d(2) = 3(6)
2d = 18
d=9
Cross Products Property
Simplify.
Divide both sides by 2.
GEOMETRY
10-1
Ratio Proportion and Similarity
TEACH! Example 3C
Solve the proportion.
(x + 3)2 = 4(9)
Cross Products Property
(x + 3)2 = 36
Simplify.
(x + 3) = 6
Find the square root of both sides.
(x + 3) = 6 or (x + 3) = –6
x = 3 or x = –9
Rewrite as two eqns.
Subtract 3 from both sides.
GEOMETRY
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Ratio Proportion and Similarity
The following table shows equivalent forms of
the Cross Products Property.
GEOMETRY
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Ratio Proportion and Similarity
Example 4: Using Properties of Proportions
Given that 18c = 24d, find the ratio of d to c in simplest
form.
18c = 24d
Divide both sides by 24c.
Simplify.
GEOMETRY
10-1
Ratio Proportion and Similarity
TEACH! Example 4
Given that 16s = 20t, find the ratio t:s in simplest form.
16s = 20t
Divide both sides by 20s.
Simplify.
GEOMETRY
10-1
Ratio Proportion and Similarity
Example 5: Problem-Solving Application
Martha is making a scale drawing of her bedroom.
1
Her rectangular room is 12
2
long.
feet wide and 15 feet
On the scale drawing, the width of her room is
5 inches. What is the length?
1
Understand the Problem
The answer will be the length of the
room on the scale drawing.
GEOMETRY
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Ratio Proportion and Similarity
Example 5 Continued
2
Make a Plan
Let x be the length of the room on the scale
drawing. Write a proportion that compares
the ratios of the width to the length.
GEOMETRY
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Ratio Proportion and Similarity
Example 5 Continued
3
Solve
5(15) = x(12.5)
75 = 12.5x
x=6
Cross Products Property
Simplify.
Divide both sides by 12.5.
The length of the room on the scale drawing is 6
inches.
GEOMETRY
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Ratio Proportion and Similarity
Example 5 Continued
4
Look Back
Check the answer in the original problem. The
ratio of the width to the length of the actual
room is 12 :15, or 5:6. The ratio of the width to
the length in the scale drawing is also 5:6. So
the ratios are equal, and the answer is correct.
GEOMETRY
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Ratio Proportion and Similarity
TEACH! Example 5
Suppose the special-effects team of “Lord of The
Ring” made a model with a height of 9.2 ft and a
width of 6 ft. What is the height of the tower
presented in the movie if the scale proportion is
1:166?
1
Understand the Problem
The answer will be the height of the tower.
GEOMETRY
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Ratio Proportion and Similarity
TEACH! Example 5 Continued
2
Make a Plan
Let x be the height of the tower. Write a
proportion that compares the ratios of
the height to the width.
GEOMETRY
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Ratio Proportion and Similarity
TEACH! Example 5 Continued
3
Solve
9.2(996) = 6(x) Cross Products Property
9163.2 = 6x
Simplify.
1527.2 = x
Divide both sides by 6.
The height of the actual tower is 1527.2 feet.
GEOMETRY
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Ratio Proportion and Similarity
TEACH! Example 5 Continued
4
Look Back
Check the answer in the original problem. The
ratio of the height to the width of the model is
9.2:6. The ratio of the height to the width of the
tower is 1527.2:996, or 9.2:6. So the ratios are
equal, and the answer is correct.
GEOMETRY
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Ratio Proportion and Similarity
Lesson Quiz
1. The ratio of the angle measures in a triangle is
1:5:6. What is the measure of each angle?
15°, 75°, 90°
Solve each proportion.
2.
X=3
3.
7 or –7
4. Given that 14a = 35b, find the ratio of a to b in
simplest form.
5. An apartment building is 90 ft tall and 55 ft wide. If a
scale model of this building is 11 in. wide, how tall is
the scale model of the building?
18 in.
GEOMETRY