geometry 10-1 ratio proportion and similarity homework: chapter 10-1 page 499 # 1-27 odds (must show...

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.