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Chapter 6 Proportions and Similarity SECTION 6.1: Proportions A RATIO is a comparison of two quantities. Ratios can be expressed as: or a:b or “a to b” Extended Ratios can be applied to 3 or More Quantities (Like Triangles) a:b:c or “a to b to c” Sample Word Problems Involving Ratios and Triangles 1.If the Ratio of the angles of a triangle is 1:2:3, then find the measure of each angle. Sample Word Problems Involving Ratios and Triangles 1. If the Ratio of the angles of a triangle is 1:2:3, then find the measure of each angle. • Solution: • The angles of this triangle have to follow this pattern – 1:2:3. Meaning that the second angle is double the first and the third is triple the first. • We also know that the angles must add up to 180°. • Therefore, represent the “multiplier” with a variable, write an equation and solve. Then substitute to find the angles. 1x + 2x + 3x = 180 6x = 180 x = 30 Therefore the angles are: 1(30)= 30°, 2(30)= 60°, and 3(30)= 90° Sample Word Problems Involving Ratios and Triangles 2. If the Ratio of the sides of a triangle is 9:8:7, and the perimeter is 144 inches. Find the measure of each side. Sample Word Problems Involving Ratios and Triangles 2. If the Ratio of the sides of a triangle is 9:8:7, and the perimeter is 144 inches. Find the measure of each side. • Since perimeter is the sum of the sides, and we know the total we can write an equation and solve as we did before. 9x + 8x + 7x = 144 24x = 144 x=6 Therefore the sides are: 9(6) = 54 in, 8(6) = 48 in, 7(6) = 42 in. Two equal Ratios form a PROPORTION Solve Each of the Following: 3 5 1. = 2. −2 2 3. −1 +1 75 = 4 5 = 5 6 Solve Each of the Following: 3 5 1. = 2. −2 2 3. −1 +1 75 = 4 5 = 5 6 1. 5x = 3(75) 5x = 225 x = 45 2. 2(4) = 5(y – 2) 8 = 5y – 10 18 = 5y 3.6 = y 3. 6(z-1) = 5(z+1) 6z – 6 = 5z + 5 z = 11 SECTION 6.2: Similar Polygons When polygons are the same shape, but different in size they are called SIMILAR, ~ Order MATTERS!!!!, We use Similarity Statements to ID Corresponding parts The 2 Polygons are SIMILAR. Write a Sim Statement, find x, y and UT. Then ID the scale factor. Sim Statement: RSTUV ~ ABCDE UT = 20.5 + 2 = 22.5 m SECTION 6.3: Similar Triangles There are 3 Rules for Proving Triangle Similarity Determine whether the triangles are similar. SECTION 6.4: Parallel Lines and Proportional Parts • = 5 • = ? 6 5 • = 4 6 • 4x = 30 • x = 7.5 u = − • = 8 • = 3 6 • 3x = 48 • x = 16 • WS = RS – RW • WS = 10 u Find x Find x • Due to the tick marks, we know that the segment labelled “x” is a midsegment. • Therefore, it is half of the segment it is parallel to. • x = (1/2) 35 • x = 17.5 u SECTION 6.5: Parts of Similar Triangles If Two Triangles are Similar, then…. •Corresponding sides are proportional, •Medians are proportional, •Altitudes are proportional, •Angle Bisectors are proportional, and •The Perimeters of the Triangles are Proportional to the sides. So, the perimeter of ∆LMN = 35 units Find x: Find x: • Solution: 20 • 7 = 24 • 20x = 168 • x = 8.4 units