Transcript 8.1 Ratio and Proportion Geometry Mrs. Spitz Spring 2005 Slide #1 Objectives/Assignment   Find and simplify the ratio of two numbers. Use proportions to solve real-life problems, such as computing the.

8.1 Ratio and Proportion
Geometry
Mrs. Spitz
Spring 2005
Slide #1
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
WS 8.1A Due end of
class
WS 8.1B Homework
Quizzes after 8.3 and
8.5 and 8.7
Slide #2
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
a:b. Because a ratio is a quotient, its
denominator cannot be zero. Ratios are
usually expressed in simplified form. For
instance, the ratio of 6:8 is usually simplified
to 3:4. (You divided by 2)
Slide #3
Ex. 1: Simplifying Ratios

a.
Simplify the ratios:
12 cm
b. 6 ft
4 cm
18 ft
Slide #4
c. 9 in.
18 in.
Ex. 1: Simplifying Ratios

a.
Simplify the ratios:
12 cm
b. 6 ft
4m
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.
Slide #5
Ex. 1: Simplifying Ratios

a.
Simplify the ratios:
12 cm
4m
12 cm
12 cm
4m
4∙100cm
12
400
Slide #6
3
100
Ex. 1: Simplifying Ratios

Simplify the ratios:
b. 6 ft
18 in
6 ft
6∙12 in
18 in
18 in.
72 in.
18 in.
Slide #7
4
1
4
Ex. 2: Using Ratios

The perimeter of
rectangle ABCD is 60
centimeters. The ratio
of AB: BC is 3:2. Find
the length and the
width of the rectangle
Slide #8
B
C
w
A
l
D
Ex. 2: Using Ratios

SOLUTION: Because
the ratio of AB:BC is
3:2, you can represent
the length of AB as 3x
and the width of BC as
2x.
Slide #9
B
C
w
A
l
D
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.
Slide #10
Ex. 3: Using Extended Ratios


The measures of the angles
in ∆JKL are in the
extended ratio 1:2:3.
Find the measures of the
angles.
Begin by sketching a
triangle. Then use the
extended ratio of 1:2:3 to
label the measures of
the angles as x°, 2x°, and
3x°.
K
2x°
x°
J
Slide #11
3x°
L
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°.
Slide #12
Ex. 4: Using Ratios

The ratios of the side
lengths of ∆DEF to
the corresponding
side lengths of ∆ABC
are 2:1. Find the
unknown lengths.
F
C
3 in.
A
D
Slide #13
8 in.
E
B
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
F
C
3 in.
A
4 in
a2 + b2 = c2
D
8 in.
E
32 + 42 = c2
9 + 16 = c2
25 = c2
5=c
Slide #14
B
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.
Slide #15
Properties of proportions
1.
CROSS PRODUCT PROPERTY. The
product of the extremes equals the product of
the means.
If
 = , then ad = bc
Slide #16
Properties of proportions
2.
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.
Slide #17
Ex. 5: Solving Proportions
4
x
4 x
4
x
=
=
=
5
7
Write the original
proportion.
7
5
Reciprocal prop.
28
5
4
Multiply each side by
4
Simplify.
Slide #18
Ex. 5: Solving Proportions
3
2
=
y+2 y
3y = 2(y+2)
3y = 2y+4
y
=
4
Write the original
proportion.
Cross Product prop.
Distributive Property
Subtract 2y from each
side.
Slide #19