AMSCO'S Geometry. New York - Rye High School

476 Ratio, Proportion, and SimilarityWhen we do not know the actual values of two or more measures that arein a given ratio, we use a variable factor to express these measures. For example,if the lengths of the sides of a triangle are in the ratio 3 : 3 : 4, we can letx be the greatest common factor of the measures of the sides. Then the measuresof the sides may be expressed as 3x,3x, and 4x. If the perimeter of the triangleis 120 centimeters, this use of the variable x allows us to write and solvean equation.3x 3x 4x 12010x 120x 12The measures of the sides of the triangle are 3(12), 3(12), and 4(12) or 36centimeters, 36 centimeters, and 48 centimeters.The Meaning of Proportion12Since the ratio 12 : 16 is equal to the ratio 3 : 4, we may write . The equation16 5 3 16 5 3 4124 is called a proportion. The proportion can also be written as12 : 16 3 : 4.DEFINITIONA proportion is an equation that states that two ratios are equal.aThe proportion b 5 dc can be written also as a : b c : d. The four numbersa, b, c, and d are the terms of the proportion. The first and fourth terms, a andd, are the extremes of the proportion, and the second and third terms, b and c,are the means.extremesCa : b c : dBmeansTheorem 12.1In a proportion, the product of the means is equal to the product of theextremes.aGivenb 5 dc with b 0 and d 0Provead bc