8.1, 2, 7: ratios, proportions and dilations objectives: be able to find and simplify the ratio of...
TRANSCRIPT
8.1, 2, 7: Ratios, Proportions and DilationsObjectives:Be able to find and simplify the ratio of two numbers.Be able to use proportions to solve real-life problems.Be able to draw dilations.
Computing RatiosIf a and b are two quantities that are measured in the same units, then the ratio of a to b is a/b. The ratio of a to b 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)
ExamplesSimplify the ratios:
121 )
4
61 )
18
cma
cm
ftb
in
3:1
4 :1
72
18
in
in
2 )
2 )
DFa
AE
BCb
DE
5
101
2
1
3
Example3) 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
B C
D
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.
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.
Example
4) The measures of the angles in a triangle are in the extended ratio of 2 : 5: 8. Find the measures of the angles and classify the triangle.2 5 8 180x x x
15 180x
12x 24 ,60 ,96
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.
a c
b d
The product of the extremes
equals the product of the means. If , then a c
ad bcb d
Example
( )(6) (4)(9)y
6 36y 6y
(4)( 4) (3)( 3)y y
4 16 3 9y y
25y
4 4 35) 6)
9 6 3 4
y
y y
Solve the proportions.
Additional Properties of Proportions
If , then (Reciprocal Property)b d
ad bca c
If , then a c a b
b d c d
If , then a c a b c d
b d b d
Geometric MeanThe geometric mean of two positive numbers a and b is the positive number x such that
ax =
xb
2x ab
x ab
Example
8) 4 and 9 9) 6 and 12
4 92 2 3 3
6
6 122 2 2 3 3
6 2
Find the geometric mean between the two numbers.
4
9
x
x
2 4 9x
6
12
x
x
2 6 12x
DilationDilation: A type of transformation (nonrigid), in which the image and preimage are similar.
Nonrigid: Image and preimage are not congruent. Therefore, length is not preserved, thus it is not an isometry.
Similar: Polygons in which their corresponding angles are congruent and the lengths of their corresponding sides are proportional.
P
R
Q
P’
Q’
R’
A dilation may be a reduction (contraction) or an enlargement (expansion).
Assignment Read Pages 457-460 and 465-467 Define: Ratio, Proportion and Geometric
Mean Pages 461-464
#12,16,20,24,28,32,36,44,45-47,52-58 even,65-66.
Pages 468-471 #10-32 even, 48-56 even.