7.1—Ratio and Proportion and Problem Solving in Geometry with Proportions Course: Geometry pre-IB Quarter: 2nd Objective: Find and simplify the ratio of two numbers. Use proportions to solve problems. SSS: MA.B.1.4.3, MA.B.2.4.1, MA.B.3.4.1, MA.C.3.4.1

Computing Ratios If a and b are two quantities that are measured in the same units, then the ratio of a to b is The ratio of a to b can also be written as a : b. Because a ratio is a quotient, the denominator (b) cannot be zero. Ratios are expressed in simplified form. For example, the ratio of 6 : 8 is usually simplified as 3 : 4.

Simplifying Ratios Simplify the ratios. 1. Convert to like units so they cancel out 2. Simplify the fraction, if possible Ratios compare two quantities; this CANNOT be reduced to 4!

Using Extended Ratios The measure of the angles in ΔJKL are in the extended ratio of 1 : 2 : 3. Find the measures of the angles. 1. Sketch the triangle (optional) 2. Triangle Sum Theorem x° + 2x° + 3x° = 180° 3. Simplify and solve for x 6x = 180 x = 30 4. Substitute value of x to find angle measures x° = 30° 2x° = 2(30) = 60° 3x° = 3(30) = 90° K 2x° x° 3x° J L

 The measures of the angles in a triangle are in the extended ratio 3 : 4 : 8. Find the measures of the angles. 36°, 48°, 96°

Using Proportions An equation that equates two ratios is a proportion. For example, if the ratio is equal to the ratio , then the following proportion can be written: The numbers a and d are the extremes while the numbers b and c are the means of the proportion. To solve a proportion you find the value of the variable. a c a c = b d b d

Properties of Proportions Cross Product Property The product of the extremes equals the product of the means. Reciprocal Property If two ratios are equal, then their reciprocals are also equal.

Solving Proportions Solve the proportions. 1. Reciprocal property 2. Solve for x 1. Cross product property 3y = 2(y + 2) 2. Simplify and solve for y 3y = 2y + 4 y = 4  by substituting 4 in the original proportion Could we have used the cross product property for the first example and the reciprocal property for the second?

 Solve the proportions.

Additional Properties of Proportions

Using Properties of Proportions Tell whether the statement is true. 1. Property of proportions 2. Apply property 3. Simplify The statement is true. 1. Property of proportions 2. Apply property The statement is false. r and 6 are switched b = 3, d = 4

Using Properties of Proportions In the diagram Find the length of 1. Substitute given values 2. Simplify 3. Solve (cross product property) 20x = 160 x = 8 The length of is 8. A 16 30 B C x 10 D E

In the diagram Find the length of 17.5 M 6 N 13 15 Q L 5 P

Geometric Mean The geometric mean of two positive numbers, a and b is the positive number x such that Solving for x, you find that the G.M., (a positive number.) For example, the geometric mean of 3 and 48: √(3)(48) = √144 = 12. To check: 3, 12, 48 - common ratio is 4

 Find the geometric mean between 5 and 125.  12 is the geometric mean between 4 and x. Find x.