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8.1 Ratio and Proportion Geometry Mr. Peebles Spring 2013
8.1 Ratio and Proportion
Geometry
Mr. Peebles
Spring 2013
Bell Ringer: Solve For X
4 x+2
8 x
=
Bell Ringer: Solve For X
4 x+2
8 x
=
Answer: x = -4
Daily Learning Target (DLT)
Friday February 22, 2013
“I can understand, apply, and remember to
simplify ratios that use geometry and real-life
word problems.”
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)
Ex. 1: Simplifying Ratios
Simplify the ratios:
a. 12 cm b. 6 ft
4 cm 18 ft
Ex. 1: Simplifying Ratios
Simplify the ratios:
a. 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.
Ex. 1: Simplifying Ratios
Simplify the ratios:
a. 12 cm
4 m
Ex. 1: Simplifying Ratios
Simplify the ratios:
a. 12 cm
4 m
12 cm 12 cm 12 3
4 m 4∙100cm 400 100
Ex. 1: Simplifying Ratios
Simplify the ratios:
b. 6 ft
18 in
Ex. 1: Simplifying Ratios
Simplify the ratios:
b. 6 ft
18 in
6 ft 6∙12 in 72 in. 4 4
18 in 18 in. 18 in. 1
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
w
lA
BC
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.
w
lA
BC
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.
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°. J
K
L
x°
2x°
3x°
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°.
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.
8 in.
3 in.
F
D E
C
A 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 8 in.
3 in.
F
D E
C
A B4 in
a2 + b2 = c2
32 + 42 = c2
9 + 16 = c2
25 = c2
5 = c
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.
Properties of proportions
1. CROSS PRODUCT PROPERTY. The
product of the extremes equals the product of
the means.
If
= , then ad = bc
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.
Ex. 5: Solving Proportions
4 x
5 7
=
Ex. 5: Solving Proportions
4 x
5 7
= Write the original
proportion.
Reciprocal prop.
Multiply each side by
4
Simplify.
x 4
7 5
= 4
4
x = 28 5
Ex. 5: Solving Proportions
3 y + 2
2 y
=
Ex. 5: Solving Proportions
3 y + 2
2 y
= Write the original
proportion.
Cross Product prop.
Distributive Property
Subtract 2y from each
side.
3y = 2(y+2)
y = 4
3y = 2y+4
Assignment
Pages 369-370 (12-21, 26-29, 62-65)
Assignment
Pages 369-370 (12-21, 26-29, 62-65)
12. 4 19. 14 62. C
13. 1-2/3 20. 7 63. G
14. 4 21. 125 Miles 64. D
15. 6-7/8 26. 13:6 65. H
16. 7.2 27. 5:4
17. 7.2 28. 4:3
18. 7.5 29. A
Closure – Exit Quiz
3 y + 4
2 y
=
