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GEOMETRY
10-1 Ratio Proportion and Similarity
Homework:
Chapter 10-1 Page 499
# 1-27 Odds
(Must Show Work)
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. (–6, 4) and (6, –2)
Solve each equation.
3. 4x + 5x + 6x = 45
4. (x – 5)2 = 81
5. Write in simplest form.
2
x = 3
x = 14 or x = –4
GEOMETRY
10-1 Ratio Proportion and Similarity
Write and simplify ratios.
Use proportions to solve problems.
Objectives
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 , of two numbers a and b can be
written as a to b, or , where b ≠ 0. For
example, the ratios 1 to 2, 1:2, and all
represent the same comparison.
:a ba
b
GEOMETRY
10-1 Ratio Proportion and Similarity
In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.
Remember!
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.
The corresponding angle pairs are congruent.
Find the ratios and compare.
18 3
24 4
AC
DF
15 3
20 4
AB
FE
12 3
16 4
BC
ED
3 with a similarity ratio , or 3:4.
4ABC FED
15 20
24
1612
18 F
E
DC
B
A
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.
A B
CD
E
F
Definition of Golden Rectangl .eABCD BCFE
CB
FE
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?
1x + 6x + 13x = 180°
20x = 180°
x = 9°
B = 6x
B = 6(9°)
B = 54°
C = 13x
C = 13(9°)
C = 117°
180oA B C
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
10-1 Ratio Proportion and Similarity
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.”
Reading Math
GEOMETRY
10-1 Ratio Proportion and Similarity
Example 3: Solving Proportions
Solve the proportion.
Cross Products Property(z – 4)2 = 5(20)
Simplify.(z – 4)2 = 100
Find the square root of both sides.(z – 4) = 10
(z – 4) = 10 or (z – 4) = –10 Rewrite as two eqns.
z = 14 or z = –6 Add 4 to both sides.
GEOMETRY
10-1 Ratio Proportion and Similarity
TEACH! Example 3A
Solve the proportion.
Cross Products Property
Simplify.
Divide both sides by 8.
3(56) = 8(x)
168 = 8x
x = 21
GEOMETRY
10-1 Ratio Proportion and Similarity
TEACH! Example 3B
Solve the proportion.
Cross Products Property
Simplify.
Divide both sides by 2.
d(2) = 3(6)
2d = 18
d = 9
GEOMETRY
10-1 Ratio Proportion and Similarity
TEACH! Example 3C
Solve the proportion.
Cross Products Property(x + 3)2 = 4(9)
Simplify.(x + 3)2 = 36
Find the square root of both sides.(x + 3) = 6
(x + 3) = 6 or (x + 3) = –6 Rewrite as two eqns.
x = 3 or x = –9 Subtract 3 from both sides.
GEOMETRY
10-1 Ratio Proportion and Similarity
The following table shows equivalent forms of the Cross Products Property.
GEOMETRY
10-1 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
11 Understand the Problem
The answer will be the length of the room on the scale drawing.
Martha is making a scale drawing of her bedroom.
Her rectangular room is feet wide and 15 feet long.
On the scale drawing, the width of her room is 5 inches. What is the length?
112
2
GEOMETRY
10-1 Ratio Proportion and Similarity
Example 5 Continued
22 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
10-1 Ratio Proportion and Similarity
Solve33
Example 5 Continued
Cross Products Property
Simplify.
Divide both sides by 12.5.
5(15) = x(12.5)
75 = 12.5x
x = 6
The length of the room on the scale drawing is 6 inches.
GEOMETRY
10-1 Ratio Proportion and Similarity
Look Back44
Example 5 Continued
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
10-1 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?
11 Understand the Problem
The answer will be the height of the tower.
GEOMETRY
10-1 Ratio Proportion and Similarity
TEACH! Example 5 Continued
22 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
10-1 Ratio Proportion and Similarity
Solve33
TEACH! Example 5 Continued
Cross Products Property
Simplify.
Divide both sides by 6.
9.2(996) = 6(x)
9163.2 = 6x
1527.2 = x
The height of the actual tower is 1527.2 feet.
GEOMETRY
10-1 Ratio Proportion and Similarity
Look Back44
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.
TEACH! Example 5 Continued
GEOMETRY
10-1 Ratio Proportion and Similarity
Lesson Quiz1. The ratio of the angle measures in a triangle is
1:5:6. What is the measure of each angle?
Solve each proportion.
2. 3.
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?
15°, 75°, 90°
X = 3 7 or –7
18 in.