8.1 Ratio and Proportion

1
8.1 Ratio and Proportion

• Geometry
• Mrs. Spitz
• Spring 2005

2
Objectives/Assignment

• Find and simplify the ratio of two numbers.
• Use proportions to solve real-life problems, such
  as computing the width of a painting.

• Chapter 8 Definitions
• Ch.8 Postulates/
• Theorems
• Pp. 461-462 1-50 all
• Quizzes after 8.3 and 8.5 and 8.7

3
Computing Ratios

• If a and b are two quantities that are measured
  in the same units, then the ratio of a to be is
  a/b. The ratio of a to be can also be written as
  ab. Because a ratio is a quotient, its
  denominator cannot be zero. Ratios are usually
  expressed in simplified form. For instance, the
  ratio of 68 is usually simplified to 34. (You
  divided by 2)

4
Ex. 1 Simplifying Ratios

• Simplify the ratios
• 12 cm b. 6 ft c. 9 in.
• 4 cm 18 ft 18 in.

5
Ex. 1 Simplifying Ratios

• Simplify the ratios
• 12 cm b. 6 ft
• 4 m 18 in
• Solution To simplify the ratios with unlike
  units, convert to like units so that the units
  divide out. Then simplify the fraction, if
  possible.

6
Ex. 1 Simplifying Ratios

• Simplify the ratios
• 12 cm
• 4 m
• 12 cm 12 cm 12 3
• 4 m 4100cm 400 100

7
Ex. 1 Simplifying Ratios

• Simplify the ratios
• b. 6 ft
• 18 in
• 6 ft 612 in 72 in. 4 4
• 18 in 18 in. 18 in. 1

8
Ex. 2 Using Ratios

• The perimeter of rectangle ABCD is 60
  centimeters. The ratio of AB BC is 32. Find
  the length and the width of the rectangle

9
Ex. 2 Using Ratios

• SOLUTION Because the ratio of ABBC is 32, you
  can represent the length of AB as 3x and the
  width of BC as 2x.

10
Solution

• Statement
• 2l 2w P
• 2(3x) 2(2x) 60
• 6x 4x 60
• 10x 60
• x 6

• Reason
• Formula for perimeter of a rectangle
• Substitute l, w and P
• Multiply
• Combine like terms
• Divide each side by 10

So, ABCD has a length of 18 centimeters and a
width of 12 cm.
11
Ex. 3 Using Extended Ratios

• The measures of the angles in ?JKL are in the
  extended ratio 123. Find the measures of the
  angles.
• Begin by sketching a triangle. Then use the
  extended ratio of 123 to label the measures of
  the angles as x, 2x, and 3x.

2x
3x
x
12
Solution

• Statement
• x 2x 3x 180
• 6x 180
• x 30

• Reason
• Triangle Sum Theorem
• Combine like terms
• Divide each side by 6

So, the angle measures are 30, 2(30) 60, and
3(30) 90.
13
Ex. 4 Using Ratios

• The ratios of the side lengths of ?DEF to the
  corresponding side lengths of ?ABC are 21. Find
  the unknown lengths.

14
Ex. 4 Using Ratios

• SOLUTION
• DE is twice AB and DE 8, so AB ½(8) 4
• Use the Pythagorean Theorem to determine what
  side BC is.
• DF is twice AC and AC 3, so DF 2(3) 6
• EF is twice BC and BC 5, so EF 2(5) or 10

4 in
a2 b2 c2 32 42 c2 9 16 c2 25 c2 5
c
15
Using Proportions

• An equation that equates two ratios is called a
  proportion. For instance, if the ratio of a/b is
  equal to the ratio c/d then the following
  proportion can be written

Means
Extremes
? ?
The numbers a and d are the extremes of the
proportions. The numbers b and c are the means
of the proportion.
16
Properties of proportions

• CROSS PRODUCT PROPERTY. The product of the
  extremes equals the product of the means.
• If
• ? ?, then ad bc

17
Properties of proportions

• RECIPROCAL PROPERTY. If two ratios are equal,
  then their reciprocals are also equal.
• If ? ?, then ?

b
a
To solve the proportion, you find the value of
the variable.
18
Ex. 5 Solving Proportions
4
5
Write the original proportion. Reciprocal
prop. Multiply each side by 4 Simplify.

x
7
4
x
7
4

4
5
28
x

5
19
Ex. 5 Solving Proportions
3
2
Write the original proportion. Cross Product
prop. Distributive Property Subtract 2y from each
side.

y 2
y
3y 2(y2)
3y 2y4
y
4