8.1 Ratio and Proportion Geometry Ms. Reser

Objectives/Assignment Find and simplify the ratio of two numbers. Use proportions to solve real-life problems, such as computing the width of a painting. Chapter 8 Definitions Ch.8 Postulates/ Theorems Pp. 461-462 #1-50 all Quizzes after 8.3 and 8.5 and 8.7

Computing Ratios If a and b are two quantities that are measured in the same units, then the ratio of a to be is a/b. The ratio of a to be 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)

Ex. 1: Simplifying Ratios Simplify the ratios: 12 cm b. 6 ft c. 9 in. 4 cm 18 ft 18 in.

Ex. 1: Simplifying Ratios Simplify the ratios: 12 cm b. 6 ft 4 m 18 in Solution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible.

Ex. 1: Simplifying Ratios Simplify the ratios: 12 cm 4 m 12 cm 12 cm 12 3 4 m 4∙100cm 400 100

Ex. 1: Simplifying Ratios Simplify the ratios: b. 6 ft 18 in 6 ft 6∙12 in 72 in. 4 4 18 in 18 in. 18 in. 1

Ex. 2: Using Ratios 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

Ex. 2: Using Ratios 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.

Solution: Statement 2l + 2w = P 2(3x) + 2(2x) = 60 6x + 4x = 60 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.

Ex. 3: Using Extended Ratios The measures of the angles in ∆JKL are in the extended ratio 1:2:3. Find the measures of the angles. Begin by sketching a triangle. Then use the extended ratio of 1:2:3 to label the measures of the angles as x°, 2x°, and 3x°. 2x° 3x° x°

Solution: Statement x°+ 2x°+ 3x° = 180° 6x = 180 x = 30 Reason Triangle Sum Theorem Combine like terms Divide each side by 6 So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

Ex. 4: Using Ratios The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths.

Ex. 4: Using Ratios SOLUTION: DE is twice AB and DE = 8, so AB = ½(8) = 4 Use the Pythagorean Theorem to determine what side BC is. DF is twice AC and AC = 3, so DF = 2(3) = 6 EF is twice BC and BC = 5, so EF = 2(5) or 10 4 in a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 5 = c

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.

Properties of proportions CROSS PRODUCT PROPERTY. The product of the extremes equals the product of the means. If  = , then ad = bc

Properties of proportions RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal. If  = , then =  b a To solve the proportion, you find the value of the variable.

Ex. 5: Solving Proportions 4 5 Write the original proportion. Reciprocal prop. Multiply each side by 4 Simplify. = x 7 4 x 7 4 = 4 5 28 x = 5

Ex. 5: Solving Proportions 3 2 Write the original proportion. Cross Product prop. Distributive Property Subtract 2y from each side. = y + 2 y 3y = 2(y+2) 3y = 2y+4 y 4 =