Study Guide and Review
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1 Choose the letter of the word or phrase that best completes each statement. a. ratio b. proportion c. means d. extremes e. similar f. scale factor g. AA Similarity Post h. SSS Similarity Theorem i. SAS Similarity Theorem j. midsegment k. dilation l. enlargement m. reduction 1. A(n) of a triangle has endpoints that are the midpoints of two sides of the triangle. j; midsegment j; midsegment 2. A(n) is a comparison of two quantities using division. a; ratio a; ratio 3. If A X and C Z, then by the. g; AA Similarity Post. 5. If then a and d are the d; extremes d; extremes 6. The ratio of the lengths of two corresponding sides of two similar polygons is the f; scale factor f; scale factor 7. A(n) is an equation stating that two ratios are equivalent. b; proportion b; proportion 8. A dilation with a scale factor of will result in a(n) m; reduction m; reduction g; AA Similarity Post. 4. A(n) is an example of a similarity transformation. k; dilation k; dilation esolutions Manual - Powered by Cognero Page 1
2 9. Solve each proportion. 11. Cross multiply. Cross multiply or 10 Cross multiply. Cross multiply esolutions Manual - Powered by Cognero Page 2
3 13. The ratio of the lengths of the three sides of a triangle is 5:8:10. If its perimeter is 276 inches, find the length of the longest side of the triangle. Just as the ratio or 5:8 is equivalent to or 5x:8x, the extended ratio can be written as 5x:8x:10x. The perimeter is 276 inches, so the sum of the lengths of the sides is Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning. Step 1: Compare corresponding angles: So the measures of the three sides are 5(12) or 60, 8 (12) or 96, and 10(12) or 120. Thus the length of the longest side is 120 inches. 120 in. 14. CARPENTRY A board that is 12 feet long must be cut into two pieces that have lengths in a ratio of 3 to 2. Find the lengths of the two pieces. Let x and 12 x be the lengths of the two pieces. Form a proportion. congruent. All corresponding angles are Step 2: Compare corresponding sides: Since, the figures are not similar. No, the polygons are not similar because the corresponding sides are not proportional. No, the polygons are not similar because the corresponding sides are not proportional. So, the length of the other piece is (12 7.2) ft or 4.8 ft. Thus the length of the two pieces are 7.2 ft and 4.8 ft. 7.2 ft and 4.8 ft esolutions Manual - Powered by Cognero Page 3
4 16. Step 1: Compare corresponding angles: Since all of the angles in the polygons are right angles, they are all congruent to each other. Therefore, corresponding angles are congruent. Step 2: Compare corresponding sides: 18. PHOTOS IF the dimensions of a photo are 2 inches by 3 inches and the dimensions of a poster are 8 inches by 12 inches, are the photo and poster similar? Explan. Yes; the ratios are the same. Yes; the ratios are the same. Since, the corresponding sides of the Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. polygons have the same scale factor, which is. Yes, the rectangles are similar because all of the corresponding angles are congruent and the corresponding sides are proportional in a 3:2 ratio. Yes, the rectangles are similar because all of the corresponding angles are congruent and the corresponding sides are proportional in a 3:2 ratio. 19. By the Reflexive Property,. 17. The two triangles in the figure below are similar. Find the value of x. So, by SAS Similarity. If two triangles are similar, then their corresponding sides are proportional. Yes, by the SAS Thm. Form a proportion and solve for x esolutions Manual - Powered by Cognero Page 4
5 20. So,. by SSS Similarity. 22. Since, then congruent alternate interior angles are formed. Therefore,. Yes, by the AA Similarity Post. Yes, by the AA Post. Yes, by the SSS Thm. 21. No, the triangles are not similar because not all corresponding angles are congruent. No, the triangles are not similar because not all corresponding angles are congruent. esolutions Manual - Powered by Cognero Page 5
6 23. TREES To estimate the height of a tree, Dave stands in the shadow of the tree so that his shadow and the tree s shadow end at the same point. Dave is 6 feet 4 inches tall and his shadow is 15 feet long. If he is standing 66 feet away from the tree, what is the height of the tree? Make a sketch of the situation. 4 feet 6 inches is equivalent to 4.5 feet. 24. Find x. In shadow problems, you can assume that the angles formed by the Sun s rays with any two objects are congruent and that the two objects form the sides of two right triangles. Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate. So, the following proportion can be written Convert all measures to inches: 81 ft = 972 inches 6 feet 4 inches = 76 inches 15 feet = 180 inches. Let x be the tree s height. Substitute and solve for x: 22.5 So, the tree is inches or 34.2 feet tall ft esolutions Manual - Powered by Cognero Page 6
7 26. STREETS Find the distance along Broadway between 37th St. and 36th St. 28. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. 27. Assuming that 38th street, 37th street and 36th street are parallel to each other, we can set up a proportion using the triangle proportionality Theorem:. The distance between 37th St. and 36th St is 220 ft. 275 ft Find the value of each variable. An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides MAPS The scale given on a map of the state of Missouri indicates that 3 inches represents 50 miles. The cities of St. Louis, Springfield, and Kansas City form a triangle. If the measurements of the lengths of the sides of this triangle on the map are 15 inches, 10 inches, and 13 inches, find the perimeter of the actual triangle formed by these cities to the nearest mile. Perimeter of the triangle on the map = = 38 inches Assume that 38 inches represents x miles. Form a proportion and solve for x. Therefore, the perimeter of the actual triangle is about 633 miles mi esolutions Manual - Powered by Cognero Page 7
8 Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. 32. GRAPHIC DESIGN Jamie wants to use a photocopier to enlarge her design for the Honors Program at her school. She sets the copier to 250%. If the original drawing was 6 inches by 9 inches, find the dimensions of the enlargement. 30. B is smaller than A, so the dilation is a reduction. The distance between the vertices at (0, 0) and (6,0) for A is 6 and between the vertices at (0, 0) and (2,0) for B is 2. So the scale factor is or. reduction; Therefore, the dimensions of the enlargement is 15 in. by 22.5 in.. 15 in. by 22.5 in. 33. BUILDING PLANS In a scale drawing of a school s floor plan, 6 inches represents 100 feet. If the distance from one end of the main hallway to the other is 175 feet, find the corresponding length in the scale drawing. Assume that x inches represents 175 feet. Form a proportion and solve for x. 31. B is larger than A, so the dilation is an enlargement. The distance between the vertices at (0, 1) and (2, 1) for A is 2. The corresponding length in the scale drawing is 10.5 inches in. The distance between the vertices at (3, 1) and (7, 1) for B is 4. So the scale factor is. enlargement; 2 esolutions Manual - Powered by Cognero Page 8
9 34. MODEL TRAINS A popular scale for model trains is the 1: 48 scale. If the actual train car had a length of 72 feet, find the corresponding length of the model in inches. Let x be the unknown length of the model train. Form a proportion and solve for x: The corresponding length of the model is 1.5 feet or 18 inches. 18 in. 35. MAPS A map of the eastern United States has a scale where 3 inches = 25 miles. If the distance on the map between Columbia, South Carolina, and Charlotte, North Carolina, is 11.5 inches what is the actual distance between the cities? Let x represent the actual distance between the cities. Form a proportion and solve for x: The actual distance between the cities is about 95.8 miles mi esolutions Manual - Powered by Cognero Page 9
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