Bellwork – 1/6/15. Unit 6: Section 6.1 Ratios, Proportions, and the Geometric Mean (Starts on Page 356)

Bellwork – 1/6/15

Unit 6: Section 6.1Ratios, Proportions, and the

Geometric Mean(Starts on Page 356)

Learning Target: Learn to solve problems by writing and solving proportions

Ratios

If a and b are two numbers or quantities, and b ≠ 0, then the ratio of a to b is ________

A ratio can also be written as ______________ We want to express ratios in their simplest form

(lowest terms).

a:b

14 72 1

Simplify ratios

Example 1:Simplify the ratio.

a. 76 cm : 8 cm b.

c. 3 yd : 36 ft d.

24

4

6

150

Use a ratio to find a Dimension

Example 2:

The perimeter of a rectangular table is 28 feet and the ratio of its length to its width is 5:2. Find the length and width of the table.

5:2Length = 5xWidth = 2x

2 L + 2 W = p

2 (5x) + 2 (2x) = 2810x + 4x = 28

14x = 28x = 2

Length 5x = 5(2) = 10 ft

Width 2x = 2(2) = 4 ft

Use extended ratios

Example 3:

The measures of the angles in ∆ BCD are in the extended ratio of 2:3:4. Find the measures of the angles.

2:3:42x:3x:4x

Sum of angles = 180

2x + 3x + 4x = 180

9x = 180

X = 20

Angle measurements = 40 , 60, 80 degrees

Proportion

• A equation that states that two ratios are equal

extreme

mean

d

c

b

a

mean

extreme

Cross product property

• In a proportion, the product of the extremes equals the product of the means

• If where and , then

0b 0dd

c

b

a

Proportions

Example 4:Solve the proportion.

a. b. c.

1610

5 x

8

5

7

2x

yy 3

2

1

1

5 (16) = 10 x

80 = 10 x

X = 8

2 ( 8) = 5 (7x)

16 = 35x

X = 16/35

3y = 2(y+1)

3y = 2y +2

Y = 2

Example 5:

As part of an environmental study, you need to estimate the number of trees in a 150 acre area. You count 270 trees in a 2 acre area and you notice that the trees seem to be evenly distributed. Estimate the total number of trees.

Set up a proportion:

40500 = 2xX = 20250 trees

• The geometric mean of two positive numbers a and b is the positive number x that satisfies .

So, and b

x

x

a

abx 2 abx

Geometric Mean

• Find the Geometric Mean of 12 and 27

abx

Geometric Mean

Homework – Geometry

Page 360: Exercises: 4, 5, 11, 13, 18 – 30 Even