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.

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!

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

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

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: 13 7

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.

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 Simplify by distributing -3x -3x Get variable on same side of equation 24 = x + 12 -12 – 12 12 = x

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

Solving Quadratic Proportions Example (x+3) = 9 4 (x+3) (x+3)(x+3) = 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

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 U T E 8 cm 3 cm U G L Y 7.5 cm 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!) 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 16 6 Triangle C 8 = 3 16 6 Triangle D 30 = 5 16 6 Triangle A 6 = 2.25 16 6 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