Geometry Chapter 7 Proportions and Similarity Chapter 7

Geometry Chapter 7: Proportions and Similarity Chapter 7 Test on Tuesday 1/26 Chapter 7 -1: Proportions

• Write ratios. • ratio • Use properties of proportions. • proportion • cross products • extremes • means Reinforcement of CA Standard 6 NS 1. 3 Use proportions to solve problems (e. g. , determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. (Key)

Prerequisite Algebra Review l Solve the following equation: Multiply both sides by the reciprocal of the fraction.

Prerequisite Algebra Review (cont. ) l Solve the following equation: 5 2

Prerequisite Algebra Review (cont. ) l Solve the following equation: 5

Ratios A comparison of two quantities using division. • b can not be = 0 • Should be written in simplified form (just like fractions) Example: The ratio of 5 and 7 can be written as 5: 7 or as the fraction and we say the ratio is “five to seven”.

Example # 2: • Gary has a bag with 4 marbles, 3 books, 5 pencils, and 2 erasers. a. What is the ratio of pencils to books? 5: 3 b. What is the ratio of marbles to the total number of items in the bag? 4: 14 2: 7 (Must be reduced!)

Write a Ratio SCHOOL The total number of students who participate in sports programs at Central High School is 500. The total number of students in the school is 2000. Find the athlete-to-student ratio to the nearest tenth. To find this ratio, divide the number of athletes by the total number of students. Answer: The athlete-to-student ratio is 1: 4.

Using Ratios Example #1 • The Perimeter of a rectangle is 60 cm. The ratio of AB: BC is 3: 2. Find the length and width of the A B rectangle. 3: 2 is in lowest terms. D AB: BC could be 3: 2, 6: 4, 9: 6, 12: 8, etc. Perimeter = l + w+ l + w AB = 3 x C = 2(l+w) BC = 2 x 60 = 3 x + 2 x + 3 x + 2 x or 60 = 2(3 x + 2 x) 60 = 10 x L = 3(6) = 18 x=6 W = 2(6) = 12

Using Ratios Example #2 • The angle measures in ABC are in the extended ratio of 2: 3: 4. Find the measure of the three angles. m A+ m B+ m C = 180 o Triangle Sum Thm. 2 x + 3 x + 4 x = 180 o 9 x = 180 o B 4 x x = 20 o m A = 40 o m B = 60 o m C = 80 o A 2 x 3 x C

Using Ratios Example #3 • The ratio of the measures of the three side lengths of a ABC are , and the perimeter is 19 m. Find the measure of each side of the triangle. Change the fractions into common denominators? Multiply everything by the common denominator.

Extended Ratios in Triangles In a triangle, the ratio of the measures of three sides is 5: 12: 13, and the perimeter is 90 centimeters. Find the measure of the shortest side of the triangle. A 13 cm B 15 cm C 38 cm D 39 cm The shortest side is 15 centimeters. The answer is B. Check Add the lengths of the sides to make sure that the perimeter is 90.

In a triangle, the ratio of the measures of three sides is 3: 4: 5, and the perimeter is 42 feet. Find the measure of the longest side of the triangle. A. 10. 5 ft B. 14 ft C. 17. 5 ft D. 37 ft

Proportions • If two ratios are equal, they can be written as a proportion. Extremes Means

Proportion Properties (The names are not important, the ideas are!!!) Cross Product Property—The product of the means equals the product of the extremes.

Proportion Practice • Which proportions are not correct? 48 = 48 24 96 32 72 48 = 48

Proportion Practice • Solve the following proportions Check your answer!

Solve Proportions by Using Cross Products B. Cross products Simplify. Add 30 to each side. Divide each side by 24. Answer: – 2

A. A. 0. 65 B. 4. 5 C. – 14. 5 D. 147

B. A. 9 B. 8. 9 C. 3 D. 1. 8 1. 2. 3. 4. A B C D

Solve Problems Using Proportions TRAINS A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model. Substitution Cross products Multiply. Divide each side by 40. Answer: The width of the model is 3. 6 inches.

Solve Problems Using Proportions Substitution Cross products Multiply. Divide each side by 40. Answer: The width of the model is 3. 6 inches.

Proportion Practice #2 • A picture of a tree is shown, the actual tree is 84 in. tall. How wide is the tree?

Two large cylindrical containers are in proportion. The height of the larger container is 25 meters with a diameter of 8 meters. The height of the smaller container is 7 meters. Find the diameter of the smaller container. A. 0. 6 m B. 2. 24 m C. 2. 52 m D. 28. 57 m

Homework Chapter 8. 1 Pg 383: 2 – 9, 12 – 26, 56 – 61, 63, 64