Transcript 7.1 Ratio and Proportion

Geometry
7.1
Ratio and Proportion
Ratio
The ratio of one number to another is the quotient
when the first number is divided by the second.
Simplest Form:
ratio of 8 to 12 is
8
2
or
12
3
If y ≠ 0, the ratio of x to y is
x
y
Ways to write a ratio
a
b
1. As a fraction
2. With a colon
a:b
3. With words
a to b
b≠0
With ratios, units must be the same.
Ratios are used to compare two numbers, such as side length or angle measure,
but the quantities being compared must be in the same units.
Example: A sheet of plywood is 0.5m long by 35 cm wide.
Find the ratio of the length to width.
Notice that the units do not affect the final ratio.
The ratio is the same no matter what measuring unit you use.
Also, a ratio has no units attached to it.
We can use ratio to find the measure of angles.
Example: Two complementary angles have measures in the ratio 2:7.
Find the measure of each angle.
Ratios can compare more than two numbers. The notation used is a:b:c…
Example: The measure of the five angles of a pentagon are in the ratio 11:7:5:4:3.
Find the measure of each angle.
Proportion: an equation that states that two ratios are equal.
a c
Examples:

a:b = c:d
b d
3 1

6 2
3:6 = 1:2
An extended proportion is an equation relating three or more ratios.
Examples:
a c e
 
b d f
3 x
y


8 16 40
x = ______
y = ________
Express the ratio AB:AD in simplest form. Be sure that the quantities are in the same units.
1.
2.
3.
4.
5.
AB
27 m
8 cm
1m
25 cm
8 cm
AD
12 m
24 cm
0.3 m
2m
8m
Find the measure of each angle.
6. Two supplementary angles have measures in the ratio 3:7.
7. The measures of the angles of a triangle are in the ratio 2:2:5.
Solve
8. The perimeter of a triangle is 48 cm and the lengths of the sides are in the ratio 3:4:5.
Find the length of each side.
Homework
pg. 243
CE #1-17 Odd
WE # 1-33 odd, calc on 33
Example
A rectangular playing field has a length of one
kilometer and a width of 520 meters. Find the
ratio of the length to the width.
Length
Width
= 1000 ÷
520
4
4
= 25
13
The ratio of length to width is 25 to 13, or 25:13
Example
Find the ratio of BC to AD.
A
BC
AD
120˚
7
3b
6b
14
60˚
D
The ratio of BC to AD:
B
7
3b
70˚
C
Example
Find the ratio of the measure of the smallest angle
of the trapezoid to that of the largest angle.
<B = 180 – 70 = 110
A
120˚
Smallest angle: <D
m<A
=
=
110˚
6b
Largest angle: <A
m<D
B
60˚
1
2
14
70˚
D
The ratio of smallest angle to largest angle: 1 to 2
C
Example
A telephone pole 7 meters tall snaps into two parts
during a wind storm. The ratio of the two parts is
3:2. Find the length of each part.
Let 3x and 2x be the two parts
3x
7
3x + 2x = 7
5x = 7
2x
7
x=
or 1.4
5
Then, 3x = 4.2 meters and 2x = 2.8 meters. These are the lengths
of the two parts of the telephone pole.
Example
The measures of the 5 interior angles of a pentagon are
in the ratio 11:7:5:4:3. Find the measure of each
angle.
Let 11x, 7x, 5x, 4x and 3x be the measures of the angles
The sum of the angles is (n-2)180, or (5-2)180.
11x + 7x + 5x + 4x + 3x = (5-2)180
30x = 540
x = 18
11(18) = 198
7(18) = 126
5(18) = 90
4(18) = 72
3(18) = 54
The angle measures are 198, 126, 90, 72 and 54.
Proportion
• an equation that states that two ratios are
equal.
• can be shown as:
c
a
=
d
b
first term
second term
or
a:b = c:d
third term
fourth term