Lesson 11.1 • Parallel and Perpendicular
Name
Period
Date
1. Find the slope of each line.
a. y 4x 7
b. y 2x 7 0
c. 3x y 4
d. 2x 3y 11
4
3
g. 1.2x 4.8y 7.3
x
i. y 2
f. x y 0
1
3
3
4
h. y x
e. y (x 1) 5
1
2
2. Plot each set of points on graph paper and connect them to form a
polygon. Classify each polygon using the most specific term that
describes it. Justify your answers by finding the slopes of the sides
of the polygon.
a. A(1, 10), B(6, 3), C(4, 17), D(9, 12)
b. P(2, 4), Q(10, 6), R(14, 10), S(6, 12)
c. W(5, 4), X(7, 7), Y(15, 5), Z(3, 8)
3. Write an equation of a line parallel to each line.
2
3
d. 0.2x 0.5y 4
a. y x 4
1
1
2
3
2
3
f. 3y 5x 10
c. x y 0
b. 3x 5y 7
e. y 2.8x
4. Write an equation of a line perpendicular to each line.
2
3
d. 0.2x 0.5y 4
a. y x 4
1
1
2
3
2
3
f. 3y 5x 10
c. x y 0
b. 3x 5y 7
e. y 2.8x
5. Write the equation of the line through the point (4, 2) and
a. Parallel to the line with equation 2x 5y 9
b. Perpendicular to the line with equation 2x 5y 9
©2007 Key Curriculum Press
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75
Lesson 11.2 • Finding the Midpoint
Name
Period
Date
1. Find the midpoint of the segment between each pair of points.
a. (6, 9) and (2, 1)
b. (7, 10) and (12, 4)
c. (11, 2) and (5, 3)
d. (4.5, 2) and (3, 6.6)
e. (0, 5) and (7, 2)
f. (13.5, 12) and (10.5, 8)
2. Find the equation of the line that satisfies each set of conditions. Write
each equation in slope-intercept form.
a. Slope 2 and y-intercept (0, 5)
2
b. Slope 3 going through the origin
c. Slope 4 going through the point (1, 7)
d. x-intercept (3, 0) and y-intercept (0, 1)
e. Goes through the points (6, 1) and (3, 7)
f. Goes through the points (5, 5) and (4, 4)
3. Write the equation of the perpendicular bisector of the line segment that
goes through each pair of points. Write the equation in point-slope form
if possible.
a. (3, 2) and (1, 4)
b. (17, 8) and (2, 5)
c. (5, 2) and (1, 5)
d. (0, 4) and (0, 6)
4. Given triangle ABC with A(1, 6), B(6, 4), and C(10, 10), write the
equation of each of the following lines in point-slope form:
a. The line containing the median through A
b. The line that is the perpendicular bisector of AB
and AC
c. The line that passes through the midpoints of BC
5. Quadrilateral ABCD has vertices A(4, 5), B(8, 9), C(13, 1), and
D(0, 14).
a. Find the most specific term to describe quadrilateral ABCD. Justify
your answer.
b. Find the midpoint of each diagonal.
c. Make an observation based on your answer to 5b.
76
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©2007 Key Curriculum Press
Lesson 11.3 • Squares, Right Triangles, and Areas
Name
Period
Date
1. Find an exact solution for each quadratic equation.
a. x 2 18
b. x 2 30 0
c. (x 5)2 14
d. (x 1)2 3 7
e. (x 1)2 3 8
f. (x 2)2 4 1
2. Calculate decimal approximations for your solutions to Exercise 1.
Round your answers to the nearest ten-thousandth. Check each answer
by substituting it into the original equation.
3. Find the area of each figure.
a.
b.
e.
c.
f.
d.
4. Find the exact length of each side of these figures from Exercise 3.
a. Figure c
b. Figure d
c. Figure f
5. On grid paper, construct a square with each area, using only
a straightedge.
a. 1 square unit
©2007 Key Curriculum Press
b. 2 square units
c. 4 square units
d. 5 square units
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77
Lesson 11.4 • The Pythagorean Theorem
Name
Period
Date
1. Find the exact solutions of each equation.
a. 52 122 a 2
b. 42 b 2 52
c. 22 52 c 2
d. 62 d 2 52
e. (25
)2 62 e 2
f. (
37 )2 f 2 (611
)2
2. Find the value of each missing side, given the lengths of the other two
x
sides. Give your answers in exact form and rounded to the nearest tenth.
a. x 7, y 8
b. y 24, z 25
c. x 13, z 19
d. x 7.8, y 13.1
e. y 31, z 50
f. x 10, y 10
z
y
3. Find the exact area of each triangle. Then give the approximate area
rounded to the nearest tenth.
a.
b.
c.
16 cm
6 mm
3 mm
9 in.
20 cm
4 in.
4. Determine whether ABC is a right triangle for each set of side lengths.
B
a
b
c
A
Show your work. Measurements are in centimeters.
a. a 7, b 8, c 11
b. a 15, b 36, c 39
c. a 14
, b 21
, c 35
d. a 213
, b 29
, c 9
5. Claudine wants to find the height of a tree at school. She measures the
C
Su
nli
gh
t
Su
nli
g
ht
shadow, and finds it to be 11 m long. When Claudine measures her own
shadow, it is 90 cm long. Claudine is 150 cm tall. How tall is the tree?
78
Discovering Algebra More Practice Your Skills
©2007 Key Curriculum Press
Lesson 11.5 • Operations with Roots
Name
Period
Date
1. Rewrite each expression with as few square root symbols as possible and
no parentheses.
a. 3
3
3
b. (35
)(22
)
c. 3
2 4
3 2 2
3
d. (32
)2
e. 3
(23
1)
615
g. 3
20
f. 5
h. 5
(1 35
)
i. 7
5 (2)(3) 5
2. Evaluate each expression.
a. (
19 )2
c. (
29 )2 42
b. (2
3 )2
d. (2
7 )2 (11
)2
3. Find the exact length of the third side of each right triangle. All
measurements are in centimeters.
a.
b.
12
c.
b
4
8
a
11
10
5
7
2
c
4. Write the equation for each parabola in general form. Use your
calculator to check that both forms give the same graph or table.
a. y (x 2
)(x 2
)
b. y (x 25
)2
5. Name the x-intercepts for each parabola in Exercise 4. Give both the
exact value and a decimal approximation to the nearest thousandth for
each x-intercept.
6. Name the vertex for each parabola in Exercise 4. Give both the exact
values and decimal approximations to the nearest thousandth for the
coordinates of each vertex.
7. Rewrite each radical expression without a coefficient.
a. 5
3
b. 2
2
c. 3
15
d. 6
10
8. Rewrite each radical expression so that the value under the radical does
not contain perfect-square factors.
a. 12
©2007 Key Curriculum Press
b. 48
c. 96
d. 500
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79
Lesson 11.6 • A Distance Formula
Name
Period
Date
1. Find the exact distances and lengths.
y
10
A
B
5
C
10
5
5
E
10
x
5
D
10
a. A to B
b. B to C
d. DE
e. E to the origin
c. CD
2. Quadrilateral MNOP has vertices M(0, 5), N(5, 3), O(7, 8), and P(2, 10).
a. Find the slope of each side.
b. Find the length of each side.
c. What kind of polygon is MNOP?
3. Triangle DEF has vertices D(4, 10), E(2, 6), and F(6, 12).
a. Find the slope of each side.
b. Find the length of each side.
c. What kind of triangle is DEF?
4. Solve each equation. Check each solution.
80
1
4
x
a. 30
x x
b. x d. 2x 8x 5
e. x 4x 3
Discovering Algebra More Practice Your Skills
c. 3x
18 x
f. 2x 3 x
©2007 Key Curriculum Press
Lesson 11.7 • Similar Triangles and Trigonometric Functions
Name
Period
Date
1. Solve each equation for x.
9
14
6
3
b. x 6
27
x
a. 2
x
16
32
d. x
2
x
32
c. 2. On a map, 2 cm represents 0.5 km.
a. What is the actual distance between two cities that are 7.25 cm apart
on the map?
b. What is the map distance between two cities that are actually
7.7 km apart?
3. Refer to the triangle at right to answer the questions.
R
q
P
a. Name the hypotenuse.
b. With respect to angle P, name the opposite side and the adjacent side.
p
r
c. With respect to angle Q, name the opposite side and the adjacent side.
p
d. What trigonometric function of angle P is the same as r?
Q
q
e. What trigonometric function of angle Q is the same as p?
f. What ratio is the same as sin Q?
4. Write a proportion and find the value of the variable for each pair of
similar triangles.
a.
7.2
w
b.
y
13
8
10.3
17
19.1
c.
6
d.
3
x
9
©2007 Key Curriculum Press
8
3
4
8
20
5
9
z
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81
Lesson 11.8 • Trigonometry
Name
Period
Date
1. Use the triangle at right for Exercise 1a–f. Fill in the correct
angle or ratio to make each statement true.
a
a. sin A □
b. cos □ c
a
d. cos1 □
e. cos A sin □
c
A
c. tan1 □ A
c
b
a
b
f. tan □ B
a
C
2. Write a trigonometric equation and solve for the indicated side length or
angle measure.
a. Find x.
x
b. Find y.
c. Find angle P.
y
114 cm
Q
40°
15°
2 cm
7.3 m
R
2
P
3 cm
3. Find the measure of each angle. Round your answer to the nearest tenth
of a degree.
3
a. sin A 4
2
b. cos B c. tan C 3
2
23
d. sin E 9
4. Find the measure of angle A for each figure. Round your answer to the
nearest tenth of a degree.
a.
4
b. A
10
c.
9
d.
A
3 10
13
6
A
14
15
A
5. Find the area of triangle KLM to the nearest 0.1 cm2. Show your work
L
including any trigonometric equations you use.
15 cm
K
82
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40°
M
©2007 Key Curriculum Press
c.
0.4
0.6
is 1; slope of BC
is 2; slope of CD
is 1;
Slope of AB
is 11 . One pair of opposite sides have
slope of AD
4
the same slope and are therefore parallel.
Quadrilateral ABCD is a trapezoid.
b.
y
0.16 Cobras win
0.4
0.6
0.4
0.6
0.4
0.096 Cobras win
0.6
0.144 Bulldogs win
0.4
0.096 Cobras win
0.6
0.144 Bulldogs win
Q(10, 6)
6
P(2, 4)
x
10
0.36 Bulldogs win
10
P(Cobras win the match) 0.16 0.096 0.096 0.352, or about 35%
R(14, 10)
S(6, 12)
LESSON 10.6 • Expected Value
1. $3.50
4
10
2. a. P(13) 3
5 0.114; P(14) 35 0.286;
15
6
P(15) 3
5 0.429; P(16) 35 0.171
4
10
15
6
b. 3
5 13 35 14 35 15 35 16 14.657,
or about 14.7 years old
c. 14.657, or about 14.7 years old
is 1; slope of QR
is 4; slope of RS
is 1;
Slope of PQ
4
4
is 4.
slope of PS
Both pairs of opposite sides have the same slope, so
opposite sides are parallel. Also, the product of the
slopes of each pair of adjacent sides equals 1, so
adjacent sides are perpendicular. Quadrilateral PQRS
is a rectangle.
c.
y
8
X(7, 7)
W(5, 4)
6
3. a.
4
5
6
7
8
9
10
3
4
5
6
7
8
9
2
3
4
5
6
7
8
1
2
3
4
5
6
7
ROLL
1
2
3
4
5
6
1
b. 8
3
d. 84
c. 6
LESSON 11.1 • Parallel and Perpendicular
1. a. 4
4
e. 3
1
i. 2
2. a.
b. 2
c. 3
1
g. 4
4
f. 9
2
d. 3
h. 1
A(1, 10)
B(6, 3)
10
Y(15, 5)
8
Z(3, 8)
is 1; slope of XY
is 3; slope of YZ
is 1;
Slope of WX
43
2
4
slope of WZ is 2.
Both pairs of opposite sides have the same slope, so
opposite sides are parallel. Quadrilateral WXYZ is a
parallelogram.
3. Answers will vary. Possible answers:
2
a. y 3x 1
b. 3x 5y 8
1
1
c. 3x 2y 1 0
d. 0.2x 0.5y 5
e. y 2.8x 1
f. 3y 5x 10
4. Answers will vary. Possible answers:
3
a. y 2x 1
b. 5x 3y 7
1
1
c. 2x 3y 1 0
d. 0.5x 0.2y 4
e. 2.8y x
f. 5y 3x 10
y
10
x
10
10
x
5. a. 2x 5y 18
b. 5x 2y 16
LESSON 11.2 • Finding the Midpoint
10
D(9, 12)
C(4, 17)
108
Discovering Algebra More Practice Your Skills / Answers
1. a. (2, 5)
d. (0.75, 2.3)
19
b. 2, 3
e. (3.5, 3.5)
1
c. 8, 2
f. (1.5, 10)
©2007 Key Curriculum Press
2. a. y 2x 5
2
b. y 3x
LESSON 11.4 • The Pythagorean Theorem
c. y 4x 3
1
d. y 3x 1
2
e. y 3x 5
f. y x
3 19
15
b. y 2 1
3 x 2
3. a. y 3 2(x 1)
3 4
c. y 2 7(x 3)
d. y 1
4. a. y 13(x 1) 6 or y 13(x 2) 7
b. y 0.7(x 2.5) 1
10
10
c. y 7 (x 2) 7 or y 7(x 5.5) 2
5. a. Quadrilateral ABCD is a parallelogram. The slope
and DC
is 1. The slope of both BC
and
of both AB
is 2. Because opposite sides have the same
AD
slope, they are parallel. Therefore quadrilateral
ABCD is a parallelogram.
5 13 4 1
is , b. The midpoint of AC
2
2
is
(4, 2.5). The midpoint of BD
0 8 14 9
(4, 2.5).
2 ,
2
c. Both diagonals have the same midpoint. In other
words, the diagonals bisect each other.
LESSON 11.3 • Squares, Right Triangles, and Areas
1. a. x 18
, or 32
b. x 30
c. x 5 14
d. x 3 or x 1
e. x 1 5
f. No real solutions
x 4.2426
b. x 5.4772
x 1.2583 or x 8.7417
x 3 or x 1
x 3.2361 or x 1.2361
No real solutions
1
3. a. 32 square units
b. 19 square units
c. 5 square units
d. 12 square units
e. 37 square units
f. 9 square units
2. a.
c.
d.
e.
f.
4. a. 45
35
, 17
, 2, 20
25
b. 18 32, 2, 10
, 26
, 10
, 40
210
c. 8 22,
18
32
, 8
22
,
18 32
5. Answers will vary. Possible answers are given.
a.
b.
c.
d.
1. a.
c.
e.
f.
a 13
b. b 3
c 29
d. d 11
e 56
2
14
f 359
2. a.
c.
d.
e.
f.
z 113
b. x 7
10.6
y 192
8
3
13.9
z 232.45
15.2
x 1539 9
19 39.2
z 200 10
2 14.1
3. a. 96 cm2
b. 2
65 in.2 16.1 in.2
9
c. 23 mm2 7.8 mm2
?
4. a. 72 82 112
113 121
No, ABC is not a right triangle.
?
b. 152 362 392
1521 1521
Yes, ABC is a right triangle.
?
c. (14
)2 (21
)2 (35
)2
14 21 35
35 35
Yes, ABC is a right triangle.
?
d. (213
)2 (29
)2 92
52 29 81
81 81
Yes, ABC is a right triangle.
5. The tree is about 18.3 m tall.
LESSON 11.5 • Operations with Roots
1. a. 33
d. 18
g. 6
5
2. a. 19
b. 610
e. 6 3
h. 5 15
b. 12
c. 45
c. 22 63
f. 2
i. 6
5 6
d. 17
3. a. a 80
b. b 14
45
cm
cm
c. c 88
2
22
cm
4. a. y x 2 2
b. y x 2 (45)x 20
5. a. x 2 1.414
b. x 25 4.472
6. a. (0, 2)
b. (25, 0) (4.472, 0)
7. a. 75
b. 8
c. 135
d. 360
8. a. 23
©2007 Key Curriculum Press
b. 43
c. 46
d. 105
Discovering Algebra More Practice Your Skills/Answers
109
LESSON 11.6 • A Distance Formula
1. a. 112 22 125
55
units
2
2
b. 6 5 61
units
c. 12 62 37
units
2
2
d. 12 4 160
410
units
e. 82 22 68
217
units
is 2; slope of NO
is 5; slope of
2. a. Slope of MN
5
2
2
5
is ; slope of MP
is .
OP
5
2
is 29
is
b. Length of MN
units; length of NO
29
units;
length
of
OP
is
29
units;
length
is 29
of MP
units.
c. Quadrilateral MNOP has four congruent sides and
each pair of adjacent sides is perpendicular, so it is
a square.
is 2; slope of EF
is 3; slope of DF
is 1.
3. a. Slope of DE
3
2
5
is 52
is
b. Length of DE
units; length of EF
is 52 units; length of DF
104 units.
c. Triangle DEF is an isosceles right triangle.
1
5
4. a. x 6
b. x 2
c. x 6
d. x 2
e. x 2 7 0.65 f. x 1
LESSON 11.7 • Similar Triangles and Trigonometric Functions
1
1. a. x 42
b. x 2
c. x 8 d. x 2
2
7.25
;
2. a. 0.5 x x 1.8125 km
y
2
;
b. 0.5 7.7 y 30.8 cm
3. a. r
b. The opposite side is p; the adjacent side is q.
c. The opposite side is q; the adjacent side is p.
110
Discovering Algebra More Practice Your Skills / Answers
d.
e.
f.
sin P
tan Q
q
r
4. Proportions will vary.
19.1
w
;
a. 7.2 10.3 w 13.4
17
5
8
;
b. 1
3 y y 278
6
3
8
3
c. x 9 ; x 6.75
8 20
z
d. 9; z 18
4 5
LESSON 11.8 • Trigonometry
a
1. a. c
e. B
a
b. B
c. b
d. B
f. A
x
;
2. a. sin 15° 114 x 29.5 cm
7.3
b. tan 40° y; y 8.7 m
2
2
c. tan1 P, or tan P ; 30°
3
3
2
2
3. a. A 48.6°
c. C 60°
b. B 45°
d. E 22.6°
4. a. A 23.6°
c. A 40.9°
b. A 57.3°
d. A 50.8°
5. Solutions may vary. One solution is:
LM
sin 40 15 ; LM 15 sin 40 9.64
KM
cos 40 15 ; KM 15 cos 40 11.49
area KLM 12(11.49)(9.64) 55.4 cm2
©2007 Key Curriculum Press