Transcript 7-1 Ratios and Proportions

7-1 Ratios and Proportions
I CAN
•Write a ratio
•Write a ratio expressing the slope of a
line.
•Solve a linear proportion
•Solve a quadratic proportion
•Use a proportion to determine if a figure
has been dilated.
A ratio compares two numbers by division. The ratio
of two numbers a and b can be written as a to b, a:b,
or
, where b ≠ 0. For example, the ratios 1 to 2,
1:2, and
all represent the same comparison.
Example:
There are 11 boys and 15 girls in class. Write
the ratio of girls to boys.
15 to 11
15:11
15
11
The order of the numbers matters!
Writing Ratios to Express Slope of a Line
In Algebra I, you learned that the slope of a line
(m) is an example of a ratio. Slope is a rate of
change and can be expressed in the following
ways:
y
x
rise
run
y2 – y1
x2 – x1
Writing Ratios to Express Slope of a Line
Write a ratio expressing the slope of
the give line.
Substitute the given values.
Simplify.
Ratios in Similar Polygons
A ratio can involve more than two numbers. For
the rectangle, the ratio of the side lengths may
be written as 3:7:3:7.
Example : Using Ratios
The ratio of the side lengths of a triangle is
4:7:5, and its perimeter is 96 cm. What is the
length of the shortest side?
Let the side lengths be 4x, 7x, and 5x.
4x + 7x + 5x = 96
16x = 96
x=6
The length of the shortest side is 4x = 4(6) = 24 cm.
A proportion is an equation stating that two
ratios are equal to each other.
In a proportion, the cross products ad and bc are equal.
Solving Linear Proportions
To solve a proportion, “CROSS MULTIPLY AND SIMPLIFY.”
Example
4 = k
10 65
10k = 260
Cross multiply
10k = 260
10
10
Simplify by dividing both sides of equation by 10
k = 26
Solving Linear Proportions
Example
3
=
4
(x + 3) (x + 8)
3(x + 8) = 4(x + 3)
Cross multiply
3x + 24 = 4x + 12
-3x
-3x
Simplify by distributing
24 = x + 12
-12
– 12
12 = x
Get variable on same side of equation
Solving Linear Proportions
Your Turn
7 =
2
3x
(x + 4)
x = -28
Solving Quadratic Proportions
Example
2y = 8
9
4y
8y2 = 72
8
8
y2 = 9
y2  9
y  3
Cross multiply
Simplify
Take the positive and negative square root of both sides
Solving Quadratic Proportions
Your Turn
14 = 2x
x
7
x  7
Solving Quadratic Proportions
Example
(x+3) = 9
4
(x+3)
(x+3)(x+3) = 36
Cross multiply
x2 + 6x + 9 = 36
-36 -36
FOIL
x2 + 6x – 27 = 0
( x – 3 )( x + 9 ) = 0
Solve quadratic equations by setting equation = 0
x -3 = 0
Use Zero Product Property to find solutions
x=3
x + 9 =0
x = -9
Factor
Solving Quadratic Proportions
Your Turn
(x – 4) = 20
5
(x – 4)
x = 14
x = -6
Solving Quadratic Proportions
Example
3 = (x – 8)
(x + 9) (3x – 8)
3(3x – 8) = (x – 8)(x + 9)
9x – 24 = x2 + 9x – 8x – 72
9x – 24 = x2 + x – 72
– 9x + 24
– 9x + 24
0 = x2 – 8x – 48
0 = (x – 12)(x + 4)
x – 12 = 0
x+4=0
x = 12 or
x=–4
Dilations and Proportions
When a figure is dilated, the pre-image and image
are proportional.
You can use proportions to find missing measures
and to check dilations!
Refer to the “Dilations as Proportions”
Worksheet in your Unit plan.
We will now work examples 1 and 2.
Dilations as Proportions
Ex) Rectangle CUTE was dilated to create rectangle
UGLY. Find the length of LY.
C
8 cm
U
G
7.5 cm
3 cm
E
U
T
Y
L
3 = 8
7.5
UG
Pre-image and image of dilated figures are proportional
3 =8
7.5 LY
Opposite sides of a rectangle are congruent.
3LY = 8(7.5)
Cross multiply
3LY = 60
LY = 20 cm
Simplify
Dilations as Proportions
Ex) Determine which of the following figures could be a dilation of the triangle on
the right (There could be more than one answer!)
A
C
B
20 in.
D
8 in.
6 in.
30 in.
2.25 in.
16 in.
3 in.
6 in.
10 in.
5 in.
Triangle A
Triangle B
Triangle C
6 = 2.25
16
6
20 = 10
16
6
8 =
16
36 = 2.25(16)?
36 = 36?
YES
20(6) = 10(16)?
120 = 160?
NO
3
6
8(6)=16(3)?
48 = 48?
YES
Triangle D
30 = 5
16
6
30(6) =16(5)?
180 = 80?
NO
Now complete #1 & 2 on
Dilations as Proportions Worksheet