Download Ratio, Proportion, and Similarity

Ratio, Proportion,
and Similarity
Unit 6: Section 7.1 and Section 7.2
The ratio of one number to another is the quotient when the first number is
divided by the second. The quotient is usually expressed in simplest form.
Ratios can be written in three forms. They can be written as a fraction, with a :, or
with the word ‘to’ between the numbers being compared and they should always
be written in simplest form
a/b a:b
a to b
For example, if you are making oatmeal for breakfast and the instructions say to
use three cups of oatmeal with two cups of water the ratio of water to oatmeal
would be 2/3, or 2:3, or 2 to 3.
Example 1
Express the following ratios in simplest form.
1) 12
20
Answers:
12 ÷ 4 = 3/5
20 ÷ 4
2) 3p
5p
p’ s cancel out
= 3/5
Practice 1: p244 #21-23
3) 4n
n2
one n cancels out
= 4/n
4) _3(x + 4)
a(x + 4)
(x + 4) cancels out
= 3/a
Example #2: Use the measurements on the given line to
find the following ratios in simplest form.
1) DI:IS
5:2
2) ST:DI
6:5
3) IT:DT
8:13
4) DI:IT
5:8
5) IT:DS
8:7
6) IS:DI:IT
12
4
10
I
2:5:6
D
Practice 2: p243 written exercises #6-14e
T
S
Example #3
5) Is the ratio of a:b always, sometimes, or never equal to b:a?
Sometimes…when a = b the ratios are equal
6) The ratio of the measures of two complementary angles is 4:5. Find the
measure of each angle.
4x + 5x = 90 (since they are complementary)
9x = 90
X = 10
The angles are 4(10) = 40° and 5 (10) = 50°
Example #4
7) The measures of the angles of a triangle are in the ratio 3:4:5. Find
measure of the largest angle.
Answer The sum of the interior angles of a triangle is 180
Let 3x, 4x, and 5x represent the measures
3x + 4x + 5x = 180
12x = 180
x = 15
3x = 45, 4x = 60, and 5x = 75
Practice 3: p244 #25
Example #5
9. a. Find the ratio of AE to BE.
10
Answer
Simplify
5x
2
x
D
A
Rewrite
60°
60°
10
E
2:x
C
b. Find the ratio of the
largest angle of △ACE
to the smallest angle
of △DBE.
90
30 Simplify
Answer
3
1
3:1
Rewrite
30°
5x
30°
B
Practice 4: p244 #1-4
A proportion is an equation stating that two ratio are equal.
a c

b d
and a:b = c:d
are equivalent forms of the same proportion. Either form can be read “a is to b
as c is to d.”
The first and third terms are called the extremes. The middle terms are the
means. The extremes are shown in red and the means are shown in blue.
Notice that a · d = b · c illustrates a property of all proportions, called the meansextremes property of proportions.
The number a is called the first term of the proportion and the numbers b, c, and
d are the second, third, and fourth terms.
When three or more ratios are equal, you can write extended proportion:
a c e
 
b d f
Properties of proportions
a c

is equivalent to:
b d
a b

a. ad = bc
b.
c d
1.
2.
a c e
 
b d f = ···, then
c.
b d

a c
a  c  e   a
  
b  d  f   b
Example 5
a 3
 to complete each statement.
Use the proportion
b 5
ab
 35
3b
b
5
1. 5a = __________
3.
__________
2.
5
3

b
a
_________
4.
b
5

a
_________
3
d.
ab cd

b
d
Practice 5:
x 4
28
1. If 7  2 , then 2x = ___________
y
2
 ___________
2. If 2x = 3y, then
x
3
6
x 4
x7
 , then
 ___________
2
3. If
7 2
7
y
x y2
x3
2
4. If
, then

 ___________
3
2
3
A
D
AD CE
Example 6: In the figure DB  EB
If CE = 2, EB = 6, and AD =3 then DB = _________
9
C
Practice 6: Using the same figure
6
If AB = 10, DB = 8, and CB = 7.5, then EB = _________
If time: P247 #9-19 odd
Homework: Practice Worksheet 7.1-7.2
E
B