7-1 Ratios and Proportions

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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.

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A ratio compares two numbers by division. The
ratio of two numbers a and b can be written as a
to b, ab, or , where b ? 0. For example, the
ratios 1 to 2, 12, and all represent the
same comparison.
Example There are 11 boys and 15 girls in
class. Write the ratio of girls to boys.
15 11
15 to 11
1511
The order of the numbers matters!
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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
y2 y1 x2 x1
rise run
y
x
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Writing Ratios to Express Slope of a Line
Write a ratio expressing the slope of the give
line.
Substitute the given values.
Simplify.
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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
3737.
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Example Using Ratios
The ratio of the side lengths of a triangle is
475, 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.
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In a proportion, the cross products ad and bc are
equal.
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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
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Solving Linear Proportions
Example 3 4 (x 3) (x 8)
3(x 8) 4(x 3)
Cross multiply
3x 24 4x 12
Simplify by distributing
-3x -3x
Get variable on same side of equation
24 x 12
-12 12
12 x
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Solving Linear Proportions
Your Turn 7 2 3x (x 4)
x -28
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Solving Quadratic Proportions
Example 2y 8 9 4y
8y2 72
Cross multiply
8
8
y2 9
Simplify
Take the positive and negative square root of
both sides
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Solving Quadratic Proportions
Your Turn 14 2x x 7
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Solving Quadratic Proportions
Example
(x3) 9 4 (x3)
(x3)(x3) 36
Cross multiply
x2 6x 9 36
FOIL
-36 -36
x2 6x 27 0
Solve quadratic equations by setting equation 0
( x 3 )( x 9 ) 0
Factor
x -3 0 x 9 0 x 3 x -9
Use Zero Product Property to find solutions
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Solving Quadratic Proportions
Your Turn
(x 4) 20 5 (x 4)
x 14 x -6
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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
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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.
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Dilations as Proportions
Ex) Rectangle CUTE was dilated to create
rectangle UGLY. Find the length of LY.
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.
Cross multiply
3LY 8(7.5)
3LY 60
LY 20 cm
Simplify
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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!)

Triangle B 20 10 16 6
Triangle C 8 3 16 6
Triangle D 30 5 16 6
Triangle A 6 2.25 16 6

8(6)16(3)? 48 48? YES
30(6) 16(5)? 180 80? NO
36 2.25(16)? 36 36? YES
20(6) 10(16)? 120 160? NO
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Now complete 1 2 on Dilations as Proportions
Worksheet
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