1. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
A blue cube is 3 times as tall as a red cube. How many red cubes can
fit into the blue cube? 27
11-1

2. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
(For help, go to Lesson 4-2.)
Write the numbers in each list without exponents.
1. 12, 22, 32, . . ., , 202, 302, . . ., 1202
Check Skills You’ll Need
11-1

3. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Solutions
1. 12, 22, 32, , 122
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
2. 102, 202, 302, , 1202
100, 400, 900, 1,600, 2,500, 3,600, 4,900,
6,400, 8,100, 10,000, 12,100, 14,400
11-1

4. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Simplify each square root. a
144 = 12
b.  –
– = – 9
Quick Check
11-1

5. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
You can use the formula d = h to estimate the distance d, in miles, to a horizon line when your eyes are h feet above the ground. Estimate the distance to the horizon seen by a lifeguard whose eyes are 20 feet above the ground.
Use the formula.
d = h
Replace h with 20.
d = (20)
Multiply.
d = 30
Find perfect squares close to 30. 25 30 36
<
Find the square root of the
closest perfect square.
25 = 5
Quick Check
The lifeguard can see about 5 miles to the horizon.
11-1

6. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Identify each number as rational or irrational. Explain.
a.  49
rational, because 49 is a perfect square
b. 0.16
rational, because it is a terminating decimal
c.  3
irrational, because 3 is not a perfect square
d. 
rational, because it is a repeating decimal
e. – 15
irrational, because 15 is not a perfect square
f. 12.69
rational, because it is a terminating decimal
g. 
Quick Check
irrational, because it neither terminates nor repeats
11-1

7. Square Roots and Irrational Numbers
PRE-ALGEBRA LESSON 11-1
Simplify each square root or estimate to the nearest integer.
1. –
Identify each number as rational or irrational.
5. The formula d = h , where h equals the height, in feet, of the viewer’s eyes, estimates the distance d, in miles, to the horizon from the viewer. Find the distance to the horizon for a person whose eyes are 6 ft above the ground.
–10 8
irrational
rational
3 mi
11-1

8. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
The Jones’ Organic Farm has 18 tomato plants and 30 string bean
plants. Farmer Jones wants every row to contain at least two
tomato plants and two bean plants. There should be as many rows
as possible, and all the rows must be the same. How should Farmer
Jones plant the rows?
6 rows, with each row containing 5 bean plants and 3 tomato plants
11-2

9. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
(For help, go to Lesson 4-2.)
Simplify.
Check Skills You’ll Need
11-2

10. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Solutions
= = 89
= = 90
11-2

11. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find c, the length of the hypotenuse.
c2 = a2 + b2
Use the Pythagorean Theorem.
Replace a with 28, and b with 21.
c2 =
c2 = 1,225
Simplify.
c = 1,225 = 35
Find the positive square root of each side.
The length of the hypotenuse is 35 cm.
Quick Check
11-2

12. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find the value of x in the triangle. Round to the nearest tenth.
a2 + b2 = c2
49 + x2 = 196
72 + x2 = 142
Use the Pythagorean Theorem.
Simplify.
Replace a with 7, b with x, and c with 14.
x =
x2 = 147
Find the positive square root of each side.
Subtract 49 from each side.
11-2

13. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
(continued)
Then use one of the two methods below to approximate
147
Method 1 Use a calculator. is
A calculator value for
147
Round to the nearest tenth.
x  
Use the table on page 778. Find the number closest to 147 in the N2 column. Then find the corresponding value in the N column. It is a little over 12.
Method 2 Use a table of square roots.
Estimate the nearest tenth.
x  
The value of x is about 12.1 in.
Quick Check
11-2

14. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
The carpentry terms span, rise, and rafter length are illustrated in the diagram. A carpenter wants to make a roof that has a span of 20 ft and a rise of 10 ft. What should the rafter length be?
c2 = a2 + b2
Use the Pythagorean Theorem.
Replace a with 10 (half the span), and b with 10.
c2 =
Square 10.
c2 =
Add.
c2 = 200
Find the positive square root.
c =
Round to the nearest tenth. c
Quick Check
The rafter length should be about 14.1 ft.
11-2

15. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Is a triangle with sides 10 cm, 24 cm, and 26 cm a right triangle?
a2 + b2 = c2
Write the equation to check.
Replace a and b with the shorter lengths and c with the longest length.
Simplify.
676 = 676
The triangle is a right triangle.
Quick Check
11-2

16. The Pythagorean Theorem
PRE-ALGEBRA LESSON 11-2
Find the missing length. Round to the
nearest tenth.
1. a = 7, b = 8, c =
2. a = 9, c = 17, b =
3. Is a triangle with sides 6.9 ft, 9.2 ft, and 11.5 ft a right triangle? Explain.
4. What is the rise of a roof if the span is 30 ft and the rafter length is
16 ft? Refer to the diagram on page 586.
10.6
14.4
yes; = 11.52
about 5.6 ft
11-2

17. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the number halfway between and 0.76.
0.772
11-3

18. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(For help, go to Lesson 1-10.)
Write the coordinates of each point.
1. A 2. D 3. G 4. J
Check Skills You’ll Need
11-3

19. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Solutions
1. A (–3, 4) 2. D (0, 3) 3. G (–4, –2) 4. J (3, –1)
11-3

20. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the distance between T(3, –2) and V(8, 3).
Use the Distance Formula.
d =
(x2 – x1)2 + (y2 – y1)2
Replace (x2, y2) with (8, 3) and (x1, y1) with (3, –2).
d =
(8 – 3)2 + (3 – (–2 ))2
Simplify.
d =
Find the exact distance. 50
d =
Round to the nearest tenth. d
The distance between T and V is about 7.1 units.
Quick Check
11-3

21. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the perimeter of WXYZ.
The points are W (–3, 2), X (–2, –1),
Y (4, 0), Z (1, 5). Use the Distance
Formula to find the side lengths.
(–2 – (–3))2 + (–1 – 2)2
WX =
1 + 9 = 10 =
Replace (x2, y2) with (–2, –1) and (x1, y1) with (–3, 2).
Simplify.
(4 – (–2))2 + (0 – (–1)2
XY = = =
Simplify. 37
Replace (x2, y2) with (4, 0) and (x1, y1) with (–2, –1).
11-3

22. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(continued) = =
(1 – 4)2 + (5 – 0)2
YZ =
Simplify.
Replace (x2, y2) with (1, 5) and (x1, y1) with (4, 0). 34
(–3 – 1)2 + (2 – 5)2
ZW =
Simplify.
Replace (x2, y2) with (–3, 2) and (x1, y1) with (1, 5). = =
25 = 5
11-3

23. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
(continued)
perimeter = 34 37 10
The perimeter is about 20.1 units.
Quick Check
11-3

24. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the midpoint of TV.
Use the Midpoint Formula.
x1 + x2 2
y1 + y2 ,
Replace (x1, y1) with (4, –3) and
(x2, y2) with (9, 2).
= ,
4 + 9 2
–3 + 2
Simplify the numerators.
= , 13 2 –1
Write the fractions in simplest form.
= , – 1 2
The coordinates of the midpoint of TV are 6 , – 1 2
Quick Check
11-3

25. Distance and Midpoint Formulas
PRE-ALGEBRA LESSON 11-3
Find the length (to the nearest tenth) and midpoint of each segment with the given endpoints.
1. A(–2, –5) and B(–3, 4) 2. D(–4, 6) and E(7, –2)
3. Find the perimeter of ABC, with coordinates A(–3, 0), B(0, 4), and C(3, 0).
9.1; (–2 , – ) 1 2
13.6; (1 , 2) 1 2 16
11-3

26. Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
Use these numbers to write as many proportions as you can: 5, 8, 15, 24 5 15 = 8 24 ,
11-4

27. Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
(For help, go to Lesson 6-2.)
Solve each proportion.
1. = 2. =
3. = 4. = 1 3 a 12 h 5 20 25 1 4 8 x 2 7 c 35
Check Skills You’ll Need
11-4

28. Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
Solutions
= =
3 • a = 1 • • h = 5 • 20
3a = h = 100
a = h = 4
= =
1 • x = 4 • • c = 2 • 35
x = c = 70
c = 10 1 3 a 12 h 5 20 25 1 4 8 x 2 7 c 35
11-4

29. Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
At a given time of day, a building of unknown height casts a shadow that is 24 feet long. At the same time of day, a post that is 8 feet tall casts a shadow that is 4 feet long. What is the height x of the building?
Since the triangles are similar,
and you know three lengths,
writing and solving a proportion
is a good strategy to use.
It is helpful to draw the triangles
as separate figures.
11-4

30. Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
(continued)
Write a proportion using the legs
of the similar right triangles.
= Write a proportion. 8 x 4 24
4x = 24(8) Write cross products.
4x = 192 Simplify.
x = 48 Divide each side by 4.
The height of the building is 48 ft.
Quick Check
11-4

31. Problem Solving Strategy: Write a Proportion
PRE-ALGEBRA LESSON 11-4
Write a proportion and solve.
1. On the blueprints for a rectangular floor, the width of the floor is 6 in. The diagonal distance across the floor is 10 in. If the width of the actual floor is 32 ft, what is the actual diagonal distance across the floor?
2. A right triangle with side lengths 3 cm, 4 cm, and 5 cm is similar to a right triangle with a 20-cm hypotenuse. Find the perimeter of the larger triangle.
3. A 6-ft-tall man standing near a geyser has a shadow 4.5 ft long.
The geyser has a shadow 15 ft long. What is the height of the geyser?
about 53 ft
48 cm
20 ft
11-4

32. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
One angle measure of a right triangle is 75 degrees. What is the measurement, in degrees, of the other acute angle of the triangle?
15 degrees
11-5

33. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
(For help, go to Lesson 11-2.)
Find the missing side of each right triangle.
1. legs: 6 m and 8 m 2. leg: 9 m; hypotenuse: 15 m
3. legs: 27 m and 36 m 4. leg: 48 m; hypotenuse: 60 m
Check Skills You’ll Need
11-5

34. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Solutions
1. c2 = a2 + b a2 + b2 = c2
c2 = b2 = 152
c2 = b2 = 225
c = = 10 m b2 = 144
b = = 12 m
3. c2 = a2 + b a2 + b2 = c2
c2 = b2 = 602
c2 = b2 = 3600
c = = 45 m b2 = 1296
b = = 36 m
11-5

35. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find the length of the hypotenuse in the triangle.
hypotenuse = leg • Use the 45°-45°-90° relationship.
y = 10 • The length of the leg is 10.
14.1 Use a calculator.
The length of the hypotenuse is about 14.1 cm.
Quick Check
11-5

36. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Patrice folds square napkins diagonally to put on a table. The side length of each napkin is 20 in. How long is the diagonal?
hypotenuse = leg • Use the 45°-45°-90° relationship.
y = 20 • The length of the leg is 20.
28.3 Use a calculator.
The diagonal length is about 28.3 in.
Quick Check
11-5

37. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find the missing lengths in the triangle.
hypotenuse = 2 • shorter leg
14 = 2 • b The length of the hypotenuse is 14.
= Divide each side by 2.
7 = b Simplify. 14 2 2b
longer leg = shorter leg • 3
a = 7 • The length of the shorter leg is 7.
a Use a calculator.
The length of the shorter leg is 7 ft. The length of the
longer leg is about 12.1 ft.
Quick Check
11-5

38. Special Right Triangles
PRE-ALGEBRA LESSON 11-5
Find each missing length.
1. Find the length of the legs of a 45°-45°-90° triangle with a hypotenuse of cm.
2. Find the length of the longer leg of a 30°-60°-90° triangle with a hypotenuse of 6 in.
3. Kit folds a bandana diagonally before tying it around her head. The side length of the bandana is 16 in. About how long is the diagonal?
4 cm
in.
about 22.6 in.
11-5

39. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
A piece of rope 68 in. long is to be cut into two pieces. How long will each piece be if one piece is cut three times longer than the other piece?
17 in. and 51 in.
11-6

40. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
(For help, go to Lesson 6-3.)
Solve each problem.
1. A 6-ft man casts an 8-ft shadow while a nearby flagpole
casts a 20-ft shadow. How tall is the flagpole?
2. When a 12-ft tall building casts a 22-ft shadow,
how long is the shadow of a nearby 14-ft tree?
Check Skills You’ll Need
11-6

41. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Solutions
= =
6 • 20 = 8 • x • 14 = 12 • x
120 = 8x = 12x
= =
x = 15 ft x = ft 6 8 x 20 12 22 14 x
120 8 8x 8
308 12
12x 12 2 3
11-6

42. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Find the sine, cosine, and tangent of A.
tan A = = =
opposite
adjacent 3 4 12 16
sin A = = =
opposite
hypotenuse 3 5 12 20
cos A = = =
adjacent
hypotenuse 4 5 16 20
Quick Check
11-6

43. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Find the trigonometric ratios of 18° using a scientific calculator or the table on page 779. Round to four decimal places.
Scientific calculator: Enter 18 and press
the key labeled SIN, COS, or TAN.
cos 18°
tan 18°
sin 18°
Table: Find 18° in the first column. Look
across to find the appropriate ratio.
Quick Check
11-6

44. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
The diagram shows a doorstop in the shape of a wedge. What is the length of the hypotenuse of the doorstop?
You know the angle and the side opposite the angle.
You want to find w, the length of the hypotenuse.
sin A = Use the sine ratio.
opposite
hypotenuse
sin 40° = Substitute 40° for the angle, 10 for
the height, and w for the hypotenuse. 10 w
w(sin 40°) = 10 Multiply each side by w.
w = Divide each side by sin 40°. 10
sin 40°
w Use a calculator.
Quick Check
The hypotenuse is about 15.6 cm long.
11-6

45. Sine, Cosine, and Tangent Ratios
PRE-ALGEBRA LESSON 11-6
Solve.
1. In ABC, AB = 5, AC = 12, and BC = 13. If A is a right angle, find the sine, cosine, and tangent of B.
2. One angle of a right triangle is 35°, and the adjacent leg is 15.
a. What is the length of the opposite leg?
b. What is the length of the hypotenuse?
3. Find the sine, cosine, and tangent of 72° using a calculator or a table. 12 13
, , 5
about 10.5
about 18.3
sin 72° ; cos 72° ; tan 72°
11-6

46. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
An airplane flies at an average speed of 275 miles per hour. How far does the airplane fly in 150 minutes?
687.5 miles
11-7

47. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
(For help, go to Lesson 2-3.)
Find each trigonometric ratio.
1. sin 45° 2. cos 32°
3. tan 18° 4. sin 68°
5. cos 88° 6. tan 84°
Check Skills You’ll Need
11-7

48. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Solutions
1. sin 45° cos 32°
3. tan 18° sin 68°
5. cos 88° tan 84°
11-7

49. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Janine is flying a kite. She lets out 30 yd of string and anchors it to the ground. She determines that the angle of elevation of the kite is 52°. What is the height h of the kite from the ground?
Draw a picture.
sin A = Choose an appropriate
trigonometric ratio.
opposite
hypotenuse
sin 52° = Substitute. h 30
30(sin 52°) = h Multiply each side by 30.
24 h Simplify.
Quick Check
The kite is about 24 yd from the ground.
11-7

50. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Quick Check
Greg wants to find the height of a tree. From his position 30 ft from the base of the tree, he sees the top of the tree at an angle of elevation of 61°. Greg’s eyes are 6 ft from the ground. How tall is the tree, to the nearest foot?
Draw a picture.
Choose an appropriate
trigonometric ratio.
opposite
adjacent
tan A =
Substitute 61 for the angle measure
and 30 for the adjacent side. h 30
tan 61° =
30(tan 61°) = h Multiply each side by 30.
54 h Use a calculator or a table.
= 60 Add 6 to account for the height
of Greg’s eyes from the ground.
The tree is about 60 ft tall.
11-7

51. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
An airplane is flying 1.5 mi above the ground. If the pilot must begin a 3° descent to an airport runway at that altitude, how far is the airplane from the beginning of the runway (in ground distance)?
Draw a picture
(not to scale).
tan 3° = Choose an appropriate trigonometric ratio.
1.5 d
d • tan 3° = 1.5 Multiply each side by d.
11-7

52. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
(continued)
= Divide each side by tan 3°.
d • tan 3°
tan 3°
1.5
d = Simplify.
1.5
tan 3°
d Use a calculator.
The airplane is about 28.6 mi from the airport.
Quick Check
11-7

53. Angles of Elevation and Depression
PRE-ALGEBRA LESSON 11-7
Solve. Round answers to the nearest unit.
1. The angle of elevation from a boat to the top of a lighthouse is 35°. The lighthouse is 96 ft tall. How far from the base of the lighthouse is the boat?
2. Ming launched a model rocket from 20 m away. The rocket traveled straight up. Ming saw it peak at an angle of 70°. If she is 1.5 m tall, how high did the rocket fly?
3. An airplane is flying 2.5 mi above the ground. If the pilot must begin a 3° descent to an airport runway at that altitude, how far is the airplane from the beginning of the runway (in ground distance)?
137 ft
57 m
48 mi
11-7