1. 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.

2. The order of the numbers matters!
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 11
15 to 11
15:11
The order of the numbers matters!

3. Writing Ratios to Express Slope of a Line
In Algebra I, you learned that the slope of a line is an example of a ratio. Slope is a rate of change and can be expressed in the following ways: y
rise
run
y2 – y1
x2 – x1 m x

4. Writing Ratios to Express Slope of a Line
Write a ratio expressing the slope of
the give line.
Substitute the given values.
Simplify.

5. Example: Writing Ratios to Express Slope of a Line
Find the slope of the line passing through the points A(2,–3) and B(–5, 10)
Answer: 7

6. A ratio can involve more than two numbers
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.

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.

8. 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.

9. Solving Linear Proportions
To solve a proportion, “CROSS MULTIPLY AND SIMPLIFY.”
Example
4 = k
10k = 260
Cross multiply
10k = 260
Simplify by dividing both sides of equation by 10
k = 26

10. Solving Linear Proportions
Example
3 =
(x + 3) (x + 8)
3(x + 8) = 4(x + 3)
Cross multiply
3x = 4x + 12
Simplify by distributing
-3x x
Get variable on same side of equation
24 = x + 12
– 12
12 = x

11. Solving Quadratic Proportions
Example
2y = 8 y
8y2 = 72
Cross multiply 8 8
y2 = 9
Simplify
Take the positive and negative square root of both sides

12. Solving Quadratic Proportions
Example
(x+3) = 9
(x+3)
(x+3)(x+3) = 36
Cross multiply
x2 + 6x + 9 = 36
FOIL
x2 + 6x – 27 = 0
Solve quadratic equations by setting equation = 0
( x – 3 )( x + 9 ) = 0
Factor
x -3 = x + 9 =0
x = x = -9
Use Zero Product Property to find solutions

13. 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 – 9x + 24
0 = x2 – 8x – 48
0 = (x – 12)(x + 4)
x – 12 = x + 4 = 0
x = or x = – 4

14. 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.

15. Dilations as Proportions
Ex) Rectangle CUTE was dilated to create rectangle UGLY. Find the length of LY. C U T E
8 cm
3 cm U G L Y
7.5 cm
3 = 8 UG
Pre-image and image of dilated figures are proportional
3 = 8 LY
Opposite sides of a rectangle are congruent.
3LY = 8(7.5)
Cross multiply
3LY = 60
LY = 20 cm
Simplify

16. 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!)
6 in.
2.25 in.
20 in.
10 in.
8 in.
3 in.
30 in.
5 in. A B C D
16 in.
6 in.
Triangle B
20 = 10
Triangle C
8 = 3
Triangle D
30 = 5
Triangle A
6 = 2.25
36 = 2.25(16)?
36 = 36?
YES
20(6) = 10(16)?
120 = 160? NO
8(6)=16(3)?
48 = 48?
YES
30(6) =16(5)?
180 = 80? NO