7.1—Ratio and Proportionand

Problem Solving in Geometry with Proportions

Computing Ratios

If a and b are two quantities that are measured in thesame units, then the ratio of a to b is

The ratio of a to b can also be written as a : b.

Because a ratio is a quotient, the denominator (b) cannot be zero.

Ratios are expressed in simplified form.For example, the ratio of 6 : 8 is usually simplified

as 3 : 4.

.b

a

Simplifying Ratios

Simplify the ratios.

1. Convert to like units so they cancel out

2. Simplify the fraction, if possible

m 4

cm 12

400

12

cm 1004

cm 12

m 4

cm 12

100

3

400

12

in. 18

ft 6

18

72

in. 18

in. 126

in. 18

ft 6

1

4

18

72

Ratios compare two quantities; this CANNOT be reduced to 4!

Using Extended RatiosThe measure of the angles in ΔJKL are in the extended ratio of 1 : 2 : 3. Find the measures of the angles.

1. Sketch the triangle (optional)

2. Triangle Sum Theoremx° + 2x° + 3x° = 180°

3. Simplify and solve for x6x = 180x = 30

4. Substitute value of x to find angle measures x° = 30° 2x° = 2(30) = 60°3x° = 3(30) = 90°

J

K

Lx° 3x°

2x°

The measures of the angles in a triangle are in the extended ratio 3 : 4 : 8. Find the measures of the angles.

36°, 48°, 96°

Using Proportions

An equation that equates two ratios is a proportion.

For example, if the ratio is equal to the ratio , then the following proportion can be written:

The numbers a and d are the extremes while the numbers b and c are the means of the proportion.

To solve a proportion you find the value of the variable.

b

a

d

c

a cb d=a cb d

Properties of Proportions

Cross Product Property

The product of the extremes equals the product of the means.

Reciprocal Property

If two ratios are equal, then their reciprocals are also equal.

. then , If bcadd

c

b

a

. then , Ifc

d

a

b

d

c

b

a

Solving Proportions

Solve the proportions.

1. Reciprocal property

2. Solve for x

1. Cross product property

3y = 2(y + 2)

2. Simplify and solve for y

3y = 2y + 4

y = 4

7

54x yy

2

2

3

5

7

4x

5

74x

5

28x

by substituting 4 in the original proportion

Could we have used the cross product property for the first example and the reciprocal property for the second?

Solve the proportions.

b

8

5

14

202.86

7b

104

5 ss

258.3

3s

Additional Properties of Proportions

. then , Ifd

b

c

a

d

c

b

a

. then , Ifd

dc

b

ba

d

c

b

a

Using Properties of ProportionsTell whether the statement is true.

1. Property of proportions

2. Apply property

3. Simplify

The statement is true.

1. Property of proportions

2. Apply property

The statement is false.

.5

3 then ,

106 If

r

prp

. then , Ifd

b

c

a

d

c

b

a

10

6

r

p

5

3

r

p

r and 6 are switched

.4

3

3

3 then ,

43 If

caca

. then , Ifd

dc

b

ba

d

c

b

a

4

4

3

3

cab = 3, d = 4

Using Properties of Proportions

In the diagram Find the length of

1. Substitute given values

2. Simplify

3. Solve (cross product property)20x = 160x = 8

The length of is 8.

.CE

AC

BD

AB .BD

BD

10

103016

x

10

2016

xD E

B C

A

3016

10x

In the diagram

Find the length of 17.5

.LP

LQ

MN

MQ

.LQ

M

N

L PQ

15

5

13

6

Geometric MeanThe geometric mean of two positive numbers, a

and b is the positive number x such that

Solving for x, you find that the G.M., (a positive number.)

For example, the geometric mean of 3 and 48:

√(3)(48) = √144 = 12.

To check: 3, 12, 48 - common ratio is 4

.b

x

x

a

,bax

Find the geometric mean between 5 and 125.

12 is the geometric mean between 4 and x. Find x.