8.1 Ratio and Proportion Geometry--Honors Mrs. Blanco.

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Presentation transcript:

8.1 Ratio and Proportion Geometry--Honors Mrs. Blanco

Investigating Ratios Put your name and each of your measurements into appropriate blank on the spreadsheet.

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

Ex. 1: Simplifying Ratios  Simplify the ratios: a. 12 cmb. 6 ftc. 9 in. 4 cm 18 in 18 in. Answers: a.b. c.

Ex. 2: Using Ratios  The perimeter of rectangle ABCD is 60 centimeters. The ratio of BC: AB is 3:2. Find the length and the width of the rectangle l=3x and w=2x

Solution: 2 l + 2w = P 2(3x) + 2(2x) = 60 6x + 4x = 60 10x = 60 x = 6 So, ABCD has a length of 18 centimeters and a width of 12 cm.

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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. x°x° 2x° 3x°

Solution: x °+ 2x°+ 3x° = 180° 6x = 180 x = 30 So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.

Ex. 4: Using Proportions  The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths. DF=6, FE=10, AB=4, CB=5

Using Proportions  An equation that equates two ratios is called a proportion.  = MeansExtremes 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

Ex. 5: Solving Proportions 4 x 5 7 = x = y y = 3y = 2(y+2) y = 4 3y = 2y+4

Ex. 4: Using Proportions  The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 5:3. Find the unknown lengths. DF=16 2/3 cm AB=6 cm 10 cm

Properties of proportions RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal. If  =, then =  b a

Section 8.2 Ex. 2: Using Properties of Proportions  In the diagram AB = BD AC CE Find the length of BD.

Solution AB = AC BD CE 16 = 20 x 10 20x = 160 x = 8 So, the length of BD is 8.