Download Lesson 3.1.1 - Solon City Schools

Lesson 3.1.1
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
3-7.
a: ≈ 30°, ≈ 40°, ≈ 110°
3-8.
a:
4
5
, y=
b: x = 3
4
5
x + 95
c: x = 6
d: x = 2
b: Obtuse scalene triangle
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.
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Lesson 3.1.2
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
20
ratios are possible. One possibility: 15
6 = 8 = 2.5
b: 25 and 10;
25
10
= 2.5 ; yes
3-20. Yes they are parallel because they have the same slope: ! 53 .
3-21. a: 6x 2 ! 8x
3-22.
b: 2x 2 + x ! 15
c: 4x 2 ! 25
d: 2x 3 ! 5x 2 ! 3x
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
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 =
42
5
b: m = 22
= 8.4
c: t = 12.5
d: x =
3
2
= 1.5
3-32. a: y = 3 ! 53 x
b: A = 7.5 sq. units; P = 8 + 34 ! 13.8
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
2
3
3-34. a: alt. int. angles
c: corresponding angles
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and ! 23 , then they are perpendicular.
b: vertical angles
d: supplementary and/or adjacent angles
2
Lesson 3.1.4
3-41. a: f = 9
b: g = 18
c: h =
70
3
3-42. 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.
3-43. a: Frank: 0.25x + 1.95 = y ; Alice: 0.40x + 1.5 = y
b: They will be 3 years old.
3-44. a: If a rectangle has base x and height 2x, then the area is 2x 2 .
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.
3-45. In theory, 3 < x < 13 but some of these lengths are not practical.
3-46. 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
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Lesson 3.2.1
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
y
3-57. See graph at right. Perimeter = 44.9 units; Area = 94 sq. units
3-58. a: ABCD ~ EVOL
x
b: RIGHT ~ RONGW
c: One possible answer: ΔTAC ~ ΔGDO
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Lesson 3.2.2
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
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Lesson 3.2.3
3-76. a: (5, –2)
b: (–4, 2)
c: (3, 3); It is the center of the figure, or the midpoint of each diagonal.
3-77. a: y =
1
2
x+2
b: A = 4 sq. units; P = 6 + 20 ! 10.47 units
c: y = !2x + 7
3-78. a: x = 51° alternate interior angles and Triangle Sum Theorem
b: x = 43° circle has 360º
c: x = 1 Pythagorean Theorem
double $200
$100
keep
3-79. a: See tree diagram at right.
b: Yes
$300
c:
1
6
,
3
6
d:
1
2
, no—the spinners are independent
e:
2
6
$600
keep $300
, because now the possible outcomes are
$100, $200, $1500, $200, $400, and $3000.
3-80. a: n = 32
$100
double
double
$1500
$3000
keep
$1500
b: m ! 14.91
3-81. Missing side length of first rectangle must be 4 m because the perimeter is 26 m. Missing
side length of second rectangle must be 9" because the area is 36 sq. in. Since angles are
equal and ratios of corresponding side lengths are equal, therefore, the rectangles are
similar. In fact, they are congruent because r = 1.
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Lesson 3.2.4
3-88. a: Scalene triangle
c: Not possible
b: Isosceles triangle
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(4x ! 12) = 180° , x = 18
b: 4.9 2 ! 3.12 = x 2 , x ! 3.79
c: x + (180° ! 51° ! 103°) + 82° = 180° , x = 72!
d: 3x ! 2 = 2x + 9 , x = 11
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Lesson 3.2.5
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
40
= 60%
b:
3-101. a: 12x 2 ! 7x ! 10
18
x
3 , x = 60
= 10
b: 16x 2 ! 8x + 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
61 ! 7.81 units
b: LD =
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 3.2.6
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
c:
3
?
=
30
75
= 40%
= 5% ; You need a total of 60 animals in the bin.
3-111. a: y = ! 12 x + 4
c: y =
2
5
x + 57
b: y = 2x ! 1
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 a
factor of 0.4. y = 7.5,!z = 9.6
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