Answer Key. 7.1 Forms of Ratios. Chapter 7 Similarity. CK- 12 Basic Geometry Concepts 1. Answers. 1. a) 4: 3. b) 5: 8. c) 6: 19. d) 6: 8: 5 2.
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1 7.1 Forms of Ratios 1. a) 4: 3 b) 5: 8 c) 6: 19 d) 6: 8: : : : : : 4: and and and CK- 12 Basic Geometry Concepts 1
2 7.2 Proportion Properties 1. x = x = 5 3. y = x = 12, y = z = gal 8. President = $800,000, VP = $600,000, Financial Officer = $400, False 10. True 11. False 12. False CK- 12 Basic Geometry Concepts 2
3 7.3 Similar Polygons and Scale Factors 1. True; all the angles are equal for all equilateral triangles. All the sides are congruent in every equilateral triangle, so the proportion of the sides is the scale factor. 2. False; the ratio of the bases can be different than the ratio of the legs. 3. False; the ratio of the lengths can be different than the ratio of the widths. 4. False; the angles of every rhombus do not have to be equal. 5. True; same reasoning as an equilateral triangle. All regular polygons are similar. 6. True; if two polygons are congruent, then they are also similar. The scale factor would be 1:1. 7. False; this is the converse of #6. Similar polygons can have a scale factor other than 1:1, meaning they would not be congruent. 8. True; all regular polygons are similar. 9. B H, I A, G T,!"!"!"!" 10.!! 11. HT = 35 CK- 12 Basic Geometry Concepts 3
4 12. IG = Ratio is!! 14. The two courts are not similar because they do not reduce to the same ratio : 9 4: 3, these ratios are not the same, so TV ratios are not the same. 16. m E = 113, m Q = !! 18. BC = CD = NP = No,!"!"!"!" 22. Yes, ABC~ NML 23. Yes, ABCD~STUV 24. Yes, EFG~ LMN 25. Yes, QRST~BCDA 26. No, m M m A and m N m C 27. No,!!"!!" 28. Yes, EFG~ MLN 29. Yes, ABDC~EFGH 30. No, we do not know any angle measures. CK- 12 Basic Geometry Concepts 4
5 7.4 AA Similarity 1. SAM~ TRI 2.!"!"!"!" 3. SM = TR = 6 5.!!! 6. ABE~ CDE because BAE DCE and ABE CDE by the Alternate Interior Angles Theorem. There is not enough information to say another other triangles are similar. 7. Possible!"!"!"!" 8. Possible AED and BEC, AEB and BEC, ABE and ABC, ECD and AED 9. AC = Yes, right angles are congruent and solving for the missing angle in each triangle, we find that the other two angles are congruent as well FE = k If an acute angle of a right triangle is congruent to an acute angle in another right triangle, then the two triangles are similar. CK- 12 Basic Geometry Concepts 5
6 14. Congruent triangles have the same shape AND size. Similar triangles only have the same shape. Congruent triangles are always similar. Similar triangles are not always congruent. 15. No only vertical angles are congruent. One angle is not enough to say the triangles are similar. 16. Yes, LNK~ JNM. 17. Yes, m IFG = 105, FIH~ GIF. 18. Yes, EB DC, so all the angles are congruent; AEB~ ADC. 19. No, there are no congruent angles. 20. Yes, vertical angles are congruent and the 55 angles are congruent; TUW~ XUV. 21. No, EG DC. CK- 12 Basic Geometry Concepts 6
7 7.5 Indirect Measurement 1. 13,000 ft ft 3. 19,400 ft ft 5. Karen, she has the longer shadow ft CK- 12 Basic Geometry Concepts 7
8 7.6 SSS Similarity 1. If all three sides in one triangle are proportional to the three sides in another, then the two triangles are similar. 2. Two triangles are similar if the corresponding sides are proportional. 3. Yes, by SSS. There are 2.2 cm in an inch, so if we were to put the larger triangle into centimeters the sides would be 15.4, 22.0, and Writing the proportions we have:!. Therefore, the side lengths are proportional.!".!!!!".! 4. No. In #3, we converted the larger triangle into centimeters. From these measurements, we can see that the larger triangle is about double the size of the smaller triangle. 5. There are 2.2 cm in an inch, so that is the scale factor. 6. ABC~ DFE 7.!"!"!"!" 8. DH = Perimeter of ABC = 36 Perimeter of DEF = 27 The ratio is 4: 3. CK- 12 Basic Geometry Concepts 8
9 10. ABC~ DBE 11. The triangles share B and!"!"!" proportional. This is SAS Similarity (in the next concept)., meaning that the two sides around B are 12. ED = !"!!!"!" 14. Yes,! =!!"!". This proportion will be valid as long as AC DE. 15. No,!!"!"!". 16. x = 6, y = 3.5 CK- 12 Basic Geometry Concepts 9
10 7.7 SAS Similarity 1. If two sides in one triangle are proportional to two sides in another and the corresponding angles are congruent, then the triangles are similar. 2. Yes, ABE CBD and!" 3. x = 3 4. x = 2 5. x = 5!"!". By SAS, ABE~ DBC. 6. Yes (we don t know which angles are what measurement, so similarity statements will vary). 7. No 8. Yes, NQP ~ 9. No 10. No Δ Δ NOM 11. Yes, cannot write a similarity statement because the vertices are not labeled. 12. No, we do not know if the lines are parallel. Cannot assume any angles are congruent. 13. No, sides don t line up. 14. No 15. Yes, cannot write a similarity statement because the vertices are not labeled. CK- 12 Basic Geometry Concepts 10
11 7.8 Triangle Proportionality : : : 3 is the ratio of the segments created by the parallel lines, 3: 5 is the ratio of the similar triangles. 8. Yes 9. No 10. Yes 11. No 12. Yes 13. No CK- 12 Basic Geometry Concepts 11
12 7.9 Parallel Lines and Transversals 1. b = y = 3 3. x = 4 4. a = 4.8, b = a = 4.5, b = 4, c = one-third of c 15. half of d CK- 12 Basic Geometry Concepts 12
13 7.10 Proportions with Angle Bisectors , , , CK- 12 Basic Geometry Concepts 13
14 7.11 Dilation , 26, !!, 3, , 10, , 3, 4 9. k!! 10. k =!! 11. k =!! CK- 12 Basic Geometry Concepts 14
15 7.12 Dilation in the Coordinate Plane 1. k =!! 2. k = 9 3. k =!! 4. A (6, 12), B (-9, 21), C (-3, -6) 5. A (9, 6), B (-3, -12), C (0, -7.5) CK- 12 Basic Geometry Concepts 15
16 6. Black triangle in graph below. 7. k = 2 Red triangle in graph below 8. Blue triangle in graph below. A (4, 8), B (48, 12), C (40, 40) 9. k = 2 CK- 12 Basic Geometry Concepts 16
17 10. OA AA AA!! OA! OA!! AB A! B! A!! B!! OA: OA = 1: 2, AB: A B = 1: 2 These ratios are the same because this is the value of the scale factor. 19. OA: OA = 1: 4, AB: A B = 1: 4 These ratios are the same because this is the value of the scale factor. CK- 12 Basic Geometry Concepts 17
18 7.13 Self-Similarity 1. Erase the middle third of each line. 2. Number of Segments Length of each Segment Total Length of the Segments Stage Stage Stage Stage Stage Stage There will be 2! segments. 4. CK- 12 Basic Geometry Concepts 18
19 5. 6. Stage 0 Stage 1 Stage 2 Stage 3 Color No Color Possible Many different flowers (roses) and vegetables (broccoli and cauliflower) are examples of fractals in nature. CK- 12 Basic Geometry Concepts 19
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