GEOMETRY: Chapter 8
8.1: Ratios and Proportions
If a and b are two numbers or quantities and
b≠0, then the ratio of a to b is a/b. The ratio
of a to b can also be written as a:b.
For example, the ratio of a side length in
triangle DEF to a side length in ABC can be
written as ½ or 1:2.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 356.
Ex. 1: Simplify the ratio.
a. 12km: 3km
b. 36 in.; 9ft
c. 100cm:2m
d. 3 ft: 30 in.
Answers:
a. 4:1
b. 1:3
c. 1:2
d. 6:5
Ex. 2:
The area of a rectangular garden is 108
square feet, and the ratio of the
length to the width is 4:3.
Find the length and width of fence
needed to enclose the garden.
Answer: length—12 feet, width—9feet
Ex. 3.
The measure of the angles of triangle RST
are in the extended ratio 2:3:4. Find the
measures of the angles.
Answers: 400, 600, 800
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 358.
Proportions: An equation that states that
two ratios are equal is called a
proportion.
The numbers b and c are the means of the
proportion. The numbers a and d are the
extremes of the proportion.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 358.
A Key Property of Proportions
1. Cross Products Property: In a
proportion, the product of the extremes
equals the product of the means.
2. Reciprocal Property—If two ratios
are equal, then their reciprocals are
also equal.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 358.
Ex.4
Solve the proportion.
8
x
a. =
24 27
2
5
b.
=
x  3 4x
Ex.4
Solve the proportion.
8
x
a. =
24 27
2
5
b.
=
x  3 4x
Answer: a. 9; b. 5
Ex.5
As part of a science project, you need to
estimate the number of blue spruce trees in a
50 acre forest. You count 36 trees in 3 acres
and notice that the trees seem to be evenly
distributed. Estimate the total number of blue
spruce trees in the forest.
Ex.5
As part of a science project, you need to
estimate the number of blue spruce trees in a
50 acre forest. You count 36 trees in 3 acres
and notice that the trees seem to be evenly
distributed. Estimate the total number of blue
spruce trees in the forest.
Answer: about 600 trees.
Geometric Mean
The geometric mean of two positive
numbers a and b is the positive number
x that satisfies a  x .
x
So, x2=ab and x 
c
ab.
Ex.6
Find the geometric mean of 36 and 54.
Ex.6
Find the geometric mean of 36 and 54.
Answer:
18 6
6.1, p. 360, #1-25 odds, 33-45 odds
(20 questions)