Ratio, Proportion, and Similarity Unit 6: Section 7.1 and Section 7.2

The ratio of one number to another is the quotient when the first number is divided by the second. The quotient is usually expressed in simplest form. Ratios can be written in three forms. They can be written as a fraction, with a :, or with the word ‘to’ between the numbers being compared and they should always be written in simplest form a/b a:b a to b For example, if you are making oatmeal for breakfast and the instructions say to use three cups of oatmeal with two cups of water the ratio of water to oatmeal would be 2/3, or 2:3, or 2 to 3. Example 1 Express the following ratios in simplest form. 1) 122) 3p3) 4n4) _3(x + 4) 20 5p n 2 a(x + 4) Answers: 12 ÷ 4 = 3/5 p’ s cancel out one n cancels out (x + 4) cancels out 20 ÷ 4 = 3/5 = 4/n = 3/a Practice 1: p244 #21-23

1) DI:IS 2) ST:DI 3) IT:DT 4) DI:IT 5) IT:DS 6) IS:DI:IT D I S T Example #2: Use the measurements on the given line to find the following ratios in simplest form. Practice 2: p243 written exercises #6-14e 5:2 8:13 5:8 2:5:6 6:5 8:7

Example #3 5) Is the ratio of a:b always, sometimes, or never equal to b:a? Sometimes…when a = b the ratios are equal 6) The ratio of the measures of two complementary angles is 4:5. Find the measure of each angle. 4x + 5x = 90 (since they are complementary) 9x = 90 X = 10 The angles are 4(10) = 40° and 5 (10) = 50°

Example #4 7) The measures of the angles of a triangle are in the ratio 3:4:5. Find measure of the largest angle. Answer The sum of the interior angles of a triangle is 180 Let 3x, 4x, and 5x represent the measures 3x + 4x + 5x = x = 180 x = 15 3x = 45, 4x = 60, and 5x = 75 Practice 3: p244 #25

A D B E 60 ° 30 ° 5x ° C 30 ° Example #5 9. a. Find the ratio of AE to BE. Answer Simplify Rewrite 2:x b. Find the ratio of the largest angle of △ ACE to the smallest angle of △ DBE. Answer Simplify Rewrite 3:1

Practice 4: p244 #1-4

A proportion is an equation stating that two ratio are equal. and a:b = c:d are equivalent forms of the same proportion. Either form can be read “a is to b as c is to d.” The first and third terms are called the extremes. The middle terms are the means. The extremes are shown in red and the means are shown in blue. Notice that a · d = b · c illustrates a property of all proportions, called the means- extremes property of proportions. The number a is called the first term of the proportion and the numbers b, c, and d are the second, third, and fourth terms. When three or more ratios are equal, you can write extended proportion:

Properties of proportions 1. is equivalent to: a.ad = bcb.c.d. 2. = ···, then Example 5 Use the proportion to complete each statement. 1.5a = __________3. __________ 2. _________4. _________ 3b

Practice 5: 1.If, then 2x = ___________ 2.If 2x = 3y, then ___________ 3.If, then ___________ 4. If, then ___________ Example 6: In the figure If CE = 2, EB = 6, and AD =3 then DB = _________ Practice 6: Using the same figure If AB = 10, DB = 8, and CB = 7.5, then EB = _________ If time: P247 #9-19 odd Homework: Practice Worksheet C E B D A