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3-5. a: D is not similar. AB = 5, BC = 4, AC =3 b: A!B! 10units, 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 4 b: MU = =x 95 41 ! + 6.40units c: One is a ratio (slope) while the other is a length (distance). 3-8. a: 5 , y 4 5 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 basesmultiplied 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 15 20 the 20 corresponds the 8. Multiple equivalent ratios aretopossible. One possibility: == 2.5 68 25 b: 25 and 10; = 2.5; yes 10 3-20. Yes they are parallel because they have the same slope: ! 3-2 1. 3-2 2. 3-2 3. a: 6x2 ! 8x b: 2x2 + x !15 c: 4x2 ! 25 3 5 . d: 2x3 ! 5x2 ! 3x x = 10° , = 61° y 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. enough evidence thatdoes a rhombus is to a rectangle. However, a rhombus not have have four right angles, so therefore there is not a: f = 9 b: g = 18 c: h = 703 3-4 Lesson 3.1.3 1. 3-4 180°factor: ! 38° !0.5; 63° =The 79°sides and 180° ! 38° ! 79° = 63°so ; corresponding angles aretoequal. 3-29. Zoom are only half as long, the side corresponding the 16b:must 2. a: a: all unmarkedtoangles thebecome same since become Upon 8, andinspection, the side corresponding the 11are must 5.5. the difference with 180° will be the same. b: It is 1:1 because it is congruent. 3-4 3-30. P(original) = 18 units and P(new) = 36 units; A(original) = 18 sq. units and A(new) = 72 sq. 3. a: Frank: 0.25x +1.95 = y; Alice: 0.40x +1.5 = y b: They will be 3 years old. units. The enlarged perimeter is 2 times greater. The enlarged area is not 2 timesgreater. Notice that the enlarged area is 4 times greater. 3-4 a: If a rectangle has base x and height 2x, then the area is 2x2. b: If a rectangle has base x 42 3 and 3y, then the perimeter is 2x + 6y . c: If a rectangle has base of 2 feet and a height 4. 3-31. a: x =height = 5 8.4 b: m = 22 c: t = 12.5 d: x = 2 = 1.5 of 3 feet, then the area is 864 sq. inches. 3 3-32. a: y = 3! 5 x 3-4 5. 3-4 3-33. 6. b: A = 7.5 sq. units; P = 8+ 34 ! In theory,5 3 < x < 13 but some of these lengths are not practical. c: y = 3+ 3 x 13.8 a: The coordinates of the image are A(–6, –4), B(10, –4), C(10, 6), D(2, 12), E(–6, 6) b: a: If the lines have Perimeters = 28 andthe 56same units;slope, areasthen = 52they and are 208parallel. sq. units b: If a line is vertical, then the slope is undefined. c: If lines have slopes 3-34. 2 3 3 and ! 2 , then they are perpendicular. a: alt. int. angles b: vertical angles c: corresponding angles d: supplementary and/or adjacent angles 3-2 1. 3-2 2. 3-2 3. a: 6x2 ! 8x 3-4 3. 3-4 4. 3-4 5. 3-4 6. c: 4x2 ! 25 d: 2x3 ! 5x2 ! 3x x = 10° , = 61° y 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 a: f = 9 3-4 1. 3-4 2. b: 2x2 + x !15 b: g = 18 c: h = 703 a: 180° ! 38° ! 63° = 79° and 180° ! 38° ! 79° = 63° ; corresponding angles are equal. b: Upon inspection, all unmarked angles are the same since the difference with 180° will be the same. a: Frank: 0.25x +1.95 = y; Alice: 0.40x +1.5 = y b: They will be 3 years old. a: If a rectangle has base x and height 2x, then the area is 2x2. b: If a rectangle has base x and height 3y, then the perimeter is 2x + 6y . c: If a rectangle has base of 2 feet and a height of 3 feet, then the area is 864 sq. inches. In theory, 3 < x < 13 but some of these lengths are not practical. a: The coordinates of the image are A(–6, –4), B(10, –4), C(10, 6), D(2, 12), E(–6, 6) b: Perimeters = 28 and 56 units; areas = 52 and 208 sq. units 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 equalangle 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 busnumber (such as 41) and then match it with each possible activity. This can be repeated for each of the possible bus numbers. 9 8 1 b: i: 12 , ii: 12 , iii: 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: ΔTAC ~ ΔGDO 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. 8 a: 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: Rotation and translation c: Rotation, dilated by factor of 2, and translation. 3-70. a: Possible b: Not possible because the sum of the measures of an obtuse and right angle is morethan 180°. c: Not possible because a triangle with sides of equal length obviously cannot have sidesof 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 theresulting 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(4x !12)= 180° , x2 = 18 2 2 b: 4.9 ! 3.1 = x , x ! 3.79 ! c: x + (180° ! 51° !103°)+ 82°= 180° , x = 72 d: 3x ! 2 = 2x + 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: 24 = 60% 40 18 b: == 60 2 3, x x 10 2 3-101. a: 12x ! 7x !10 b: 16x ! 8x +1 b: x = ! 5 9 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: ! 5 6 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 3.2.4 3-108. x = 137°,!y = 76° 3-109. h = 5 feet; perimeter ≈ 24.2 feet 28 3-110. a: =? 5 ; 2There are 70 animals in the bin. b: 3 c: ? = 5%; 13+17 30 = = 40% 75 22+8+13+15+17 You need a total of 60 animals in the bin. 3-111. a: 1 y = ! 2 x + 4 b: y = 2x !1 c: y = 2 x 7 5 5 + d: C = 15+ 7(t !1)= 8+ 7t 3-112. ≈ 13.2 miles 3-113. Possible response: Rotate WXYZ clockwise, translate it to the left, and dilate it by afactor of 0.4. y = 7.5,!z = 9.6 Selected Answers