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3 - 5. a: D is not similar. AB = 5, BC = 4, AC = 3 b: A! B! = 100 = 10 units , B! C != 8 units, and A! C! = 6 units. c: A = 24 sq. units; P = 24 units 3 - 6. a: x = 18 b: x = 3 c: x = 6 d: x = 2 3 - 7. a: ≈ 30°, ≈ 40°, ≈ 110° b: Obtuse scalene triangle 3 - 8. a: 45 , y = 45 x + 95 b: MU = 41! 6.40 units c: One is a ratio (slope) while the other is a length (distance). 3 - 9. a: triangle inequality b: Pythagorean Theorem c: base angles not equal 3 - 10. a: If a shape is an equilateral triangle, then it has 120° rotation symmetry. b: If a shape is a rectangle, then the shape is a parallelogram. c: If a shape is a trapezoid, then the area of the shape is half the sum of its bases multiplied by its height.
3 - 18. Result should be 12 units tall and 16 units wide. 3 - 19. a: The 15 corresponds to the 6, while the 20 corresponds to the 8. Multiple equivalent ratios are possible. One possibility: 156 = 208 = 2. b: 25 and 10; 2510 = 2.5 ; yes 3 - 20. Yes they are parallel because they have the same slope:! 53. 3 - 21. a: 6 x^2! 8 x b: 2 x^2 + x! 15 c: 4 x^2! 25 d: 2 x^3! 5 x^2! 3 x 3 - 22. x = 10 ° , y = 61 ° 3 - 23. No, this is not convincing. While the facts are each correct, the conclusion is not based on the facts. As stated in Fact #2, a square is a rectangle because it has four right angles. However, a rhombus does not have to have four right angles, so therefore there is not enough evidence that a rhombus is a rectangle.
Lesson 3.1.
3 - 29. a: Zoom factor: 0.5; The sides are only half as long, so the side corresponding to the 16 must become 8, and the side corresponding to the 11 must become 5.5. b: It is 1:1 because it is congruent. 3 - 30. P(original) = 18 units and P(new) = 36 units; A(original) = 18 sq. units and A(new) = 72 sq. units. The enlarged perimeter is 2 times greater. The enlarged area is not 2 times greater. Notice that the enlarged area is 4 times greater. 3 - 31. a: x = 425 = 8.4 b: m = 22 c: t = 12.5 d: x = 32 = 1. 3 - 32. a: y = 3! 53 x b: A = 7.5 sq. units; P = 8 + 34! 13. c: y = 3 + 53 x 3 - 33. a: If the lines have the same slope, then they are parallel. b: If a line is vertical, then the slope is undefined. c: If lines have slopes 23 and! 32 , then they are perpendicular. 3 - 34. a: alt. int. angles b: vertical angles c: corresponding angles d: supplementary and/or adjacent angles
3 - 53. a: Yes, since all trees are green and the oak is a tree. b: No, only trees must be green according to the statement. c: No, the second statement reverses the first. 3 - 54. a: Yes, AA ~. Dilate from right vertex. b: Yes, AA ~ since all angles are 60°. c: Yes, zoom factor of 2.5; translate so that one pair of corresponding vertices coincide, rotate so that rays coincide, and dilate. d: No, since corresponding angles are not equal. Note that you cannot apply zoom factor to angles. 3 - 55. a: One strategy: Translate one so that the centers coincide. Then dilate so that the radius is the same as the other circle. b: Equilateral triangles, which from part (b) of 3-54 were similar because they have equal angle measures. Squares or other regular polygons are also always similar. 3 - 56. a: There are 12 combinations. One way to systematically list them all is to list a bus number (such as 41) and then match it with each possible activity. This can be repeated for each of the possible bus numbers. b: i : 9 12 ,^ ii:^ 8 12 ,^ iii :^ 1 12 3 - 57. See graph at right. Perimeter = 44.9 units; Area = 94 sq. units 3 - 58. a: ABCD ~ EVOL b: RIGHT ~ RONGW c: One possible answer: Δ TAC ~ Δ GDO x y
3 - 65. a: x = 20 mm b: w = 91 mm 3 - 66. a: Impossible: can be rejected using Triangle Inequality or Pythagorean Theorem. b: Possible c: Impossible: rejected because the sum of the angles is 179°. 3 - 67. a: 8 12 b:^ 4 8 3 - 68. This reasoning is incorrect. Rewrite “it is raining” in the lower left oval, and “Andrea’s flowers must be closed up” in the right oval. 3 - 69. a: Reflection, rotation, and translation b: Not enough information to determine similarity. c: Rotation, dilated by factor of 2, and translation. d: Rotation, reflection, and reduced by zoom factor of 0.5 (or translation and reflection instead of rotation and reflection). 3 - 70. a: Possible b: Not possible because the sum of the measures of an obtuse and right angle is more than 180°. c: Not possible because a triangle with sides of equal length obviously cannot have sides of different lengths. d: Possible
3 - 88. a: Scalene triangle b: Isosceles triangle c: Not possible d: Equilateral triangle 3 - 89. a: The two equations should have the same slope but a different y - intercept. This forces the lines to be parallel and not intersect. b: When solving a system of equations that has no solution, the equations combine to create an impossible equality, such as 3 = 0. Another special case occurs when the resulting equality is always true, such as 2 = 2. This is the result when the two lines coincide, creating infinite points of intersection. 3 - 90. a: Not similar, interior angles are different. b: Must be similar by AA ~. c: Similar, all side lengths have the same ratio. 3 - 91. Perimeter = 10 + 10 + 4 + 3 + 4 + 3 + 4 = 38 units, height of triangle = 8 units, area = 60 square units. 3 - 92. This reasoning is correct. 3 - 93. a: 3 ( 4 x! 12 ) = 180 ° , x = 18 b: 4. (^2)! 3.1 (^2) = x 2 , x^!^ 3. c: x + ( 180 °! 51 °! 103 °) + 82 ° = 180 ° , x^ =^72 ! d: 3 x^!^2 =^2 x^ +^9 , x^ =^11
3 - 99. a: SSS ~ and SAS ~ (if you show that the triangles are right triangles) b: AA ~ and SAS ~ c: None since there is not enough information. 3 - 100. a: 2440 = 60% b: (^18) x = 103 , x = 60 3 - 101. a: 12 x^2! 7 x! 10 b: 16 x^2! 8 x + 1 b: x^ =^!^59 c: x = 3 3 - 102.! y = 48!^ because of vertical angles;! z = 48!^ because of reflection of! y or because of angle of incidence = angle of reflection with! x. 3 - 103. a:! (^56) b: LD = 61! 7.81units c: Calculate ∆ x and ∆ y by determining the difference in the corresponding coordinates. 3 - 104. Original: A = 135 sq. units, P = 48 units; New: A = 15 sq. units, P = 16 units
Lesson 3.2.
3 - 108. x = 137 °,! y = 76 ° 3 - 109. h = 5 feet; perimeter ≈ 24.2 feet 3 - 110. a: (^28)? = 25 ; There are 70 animals in the bin. b: 13 + 17 22 + 8 + 13 + 15 + 17 =^ 30 75 =^ 40% c: (^3)? = 5% ; You need a total of 60 animals in the bin. 3 - 111. a: y =! 12 x + 4 b: y = 2 x! 1 c: y = 25 x + 75 d: C = 15 + 7 ( t! 1 ) = 8 + 7 t 3 - 112. ≈ 13.2 miles 3 - 113. Possible response: Rotate WXYZ clockwise, translate it to the left, and dilate it by a factor of 0.4. y = 7.5,! z = 9.
