7-1 Ratio and Proportion
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1 7-1 Ratio and Proportion Ratio 1) Find the slope of line m provided that points and lie on m. 2) The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle? Proportion Cross products Solve each proportion. 3) 4) 5) Properties of Proportions The proportion is equivalent to the following. Numeric Example 6) Given that, find the ratio in simplest form. 7) During the filming of The Lord of the Rings, the special-effects team built a model of Sauron s tower with a height of 8m and a width of 6m. If the width of the full-size tower is 996m, what is its height?
2 8) ART An artist in Portland, Oregon, makes bronze sculptures of dogs. The ratio of the height of a sculpture to the actual height of the dog is 2:3. If the height of the sculpture is 14 inches, find the height of the dog. 9) SCHOOL The ratio of male students to female students in the drama club at Campbell High School is 3:4. If the number of the male students in the club is 18, what is the number of female students? 10) The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. What is the length of each side of the triangle? 11) The ratio of the measures of the three angles of a triangle is 2:5:3. What is the measure of each angle? 12) The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 inches. What is the length of each side of the triangle? 13) The ratio of the measures of the three angles of a triangle is 6:9:10. What is the measure of each angle?
3 7-2 Ratios in Similar Polygons Similar (~) Determine if the following shapes are similar. 1) 2) 3) Similar polygons 4) Identify the pairs of congruent angles and corresponding sides for and. 5) Identify the pairs of congruent angles and corresponding sides. Similarity ratio 6) Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. Find the value of x.
4 7) Determine if If so, write the similarity ratio and similarity statement. 8) A boxcar has the dimensions shown. A model of the boxcar is 1.25 in. wide. Find the length of the model to the nearest inch. Think and Discuss 9) If you combine the symbol for similarity with the equal sign, what symbol is formed? 10) The similarity ratio of rectangle ABCD to rectangle EFGH is. How do the side lengths of rectangle ABCD compare to the corresponding side lengths of rectangle EFGH? 11) What shape(s) are always similar?
5 Angle-Angle Similarity (AA ~) 7-3 Triangle Similarity: AA, SSS, and SAS Side-Side-Side Similarity (SSS ~) Side-Angle-Side Similarity (SAS ~) Refer to the diagram and complete each statement. 1) KLM ~ 2) N 5) EFD ~ 6) F 3) MK KM NJ EF FD EG 4) 7) 8) NJ NL KL EH HG FD Given JKL ~ RST, complete the following statements: 9) K 10) T 11) ~ 12) KLJ ~ 13) SRT JK KL ST RT Determine whether each pair of triangles is similar. Justify your answers.
6 14) 15) 16) 17) Identify the similar triangles, find x, and the measures of the indicated sides. 18) 19) 20) 21)
7 Triangle Proportionality Theorem 7-4 Applying Properties of Similar Triangles 1) 2) Converse of the Triangle Proportionality Theorem Determine whether BC // DE. 3) AD 15, DB 12, AE 10,and EC 8 4) BD 9, BA 27, and CE is one third of EA 5) AE 30, AC 45, and AD is twice DB
8 Two-Transversal Proportionality AC BD DF A B Other proportions that also work: C D CE AC DF AE BF CE E F EXAMPLES: 9) Find x and y. 10) Find x and y. Triangle Angle Bisector Theorem b a 11) Find x. c Others that work: a d b and c d a x b a c 8 x + 4 d
9 7-5 Using Proportional Relationships Indirect measurement 1) A student who is 5 ft. 6in. tall measured his shadow and found that it was 5 ft. long. The student then measures the shadow of a nearby flagpole and finds that it is 14 ft. 2in. long. How tall is the flagpole? Draw a picture. Scale drawing Scale 2) Find the actual distance between City Hall and El Centro College. 3) The rectangular central chamber of the Lincoln Memorial in Washington, D.C. is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in: 20 ft.
10 Proportional Perimeters and Areas Theorem 4),, and. If and for, find the perimeter and area of Draw a picture. Think and discuss 5) Explain how to find the actual distance between two cities 5.5 in. apart on a map that has a scale of 1 in: 25 mi.
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The common ratio in (ii) is called the scaled-factor. An example of two similar triangles is shown in Figure 47.1. Figure 47.1
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