square roots and irrational numbers

11-1

Square Roots and Irrational NumbersSquare Roots and Irrational NumbersPRE-ALGEBRA LESSON 11-1PRE-ALGEBRA LESSON 11-1

A blue cube is 3 times as tall as a red cube. How many red cubes canfit into the blue cube?

27


(For help, go to Lesson 4-2.)

Write the numbers in each list without exponents.

1. 12, 22, 32, . . ., 122 2. 102, 202, 302, . . ., 1202

Check Skills You’ll Need

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

b. – 81

Square Roots and Irrational NumbersSquare Roots and Irrational Numbers

Simplify each square root.

PRE-ALGEBRA LESSON 11-1PRE-ALGEBRA LESSON 11-1

a. 144

144 = 12

– 81 = – 9

Quick Check

11-1


You can use the formula d = 1.5h 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.

The lifeguard can see about 5 miles to the horizon.

Find the square root of the closest perfect square.

25 = 5

Use the formula.d = 1.5h

Replace h with 20.d = 1.5(20)

Multiply.d = 30

Find perfect squares close to 30.25 30 36< <

Quick Check

11-1

c. 3

a. 49

Square Roots and Irrational NumbersSquare Roots and Irrational Numbers

Identify each number as rational or irrational. Explain.


rational, because 49 is a perfect square

rational, because it is a terminating decimal

irrational, because 3 is not a perfect square

rational, because it is a repeating decimal

irrational, because 15 is not a perfect square

rational, because it is a terminating decimal

irrational, because it neither terminates nor repeats

e. – 15

g. 0.1234567 . . .

f. 12.69

d. 0.3333 . . .

b. 0.16

Quick Check

11-1

Simplify each square root or estimate to the nearest integer.

1. – 100 2. 57

Identify each number as rational or irrational.

3. 48 4. 0.0125

5. The formula d = 1.5h , 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.


irrational

–10 8

rational

3 mi

11-1

11-2

The Pythagorean TheoremThe Pythagorean TheoremPRE-ALGEBRA LESSON 11-2PRE-ALGEBRA LESSON 11-2

The Jones’ Organic Farm has 18 tomato plants and 30 string beanplants. Farmer Jones wants every row to contain at least twotomato plants and two bean plants. There should be as many rowsas possible, and all the rows must be the same. How should FarmerJones plant the rows?

6 rows, with each row containing 5 bean plants and 3 tomato plants



Simplify.

1. 42 + 62 2. 52 + 82

3. 72 + 92 4. 92 + 32


11-2


Solutions

1. 42 + 62 2. 52 + 82 16 + 36 = 52 25 + 64 = 89

3. 72 + 92 4. 92 + 32 49 + 81 = 130 81 + 9 = 90

11-2

The Pythagorean TheoremThe Pythagorean Theorem

Find c, the length of the hypotenuse.


c2 = a2 + b2 Use the Pythagorean Theorem.

c2 = 1,225 Simplify.

c = 1,225 = 35 Find the positive square root of each side.

The length of the hypotenuse is 35 cm.

Replace a with 28, and b with 21.c2 = 282 + 212

Quick Check

11-2


Find the value of x in the triangle.

Round to the nearest tenth.


x = 147

x2 = 147

Find the positive square root of each side.

Subtract 49 from each side.

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.

11-2

Then use one of the two methods below to approximate .147


(continued)


The value of x is about 12.1 in.

Estimate the nearest tenth.x 12.1

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.

Method 1 Use a calculator.

is 12.124356.A calculator value for 147

Round to the nearest tenth.x 12.1

Quick Check

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?


The rafter length should be about 14.1 ft.

c2 = a2 + b2 Use the Pythagorean Theorem.

Round to the nearest tenth.c 14.1

Find the positive square root.c = 200

Add.c2 = 200

Square 10.c2 = 100 + 100

Replace a with 10 (half the span), and b with 10.c2 = 102 + 102

Quick Check

11-2


Is a triangle with sides 10 cm, 24 cm, and 26 cm

a right triangle?


The triangle is a right triangle.

Simplify.100 + 576 676

Replace a and b with the shorter lengths and c with the longest length.

102 + 242 262

a2 + b2 = c2 Write the equation to check.

676 = 676

Quick Check

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 is16 ft? Refer to the diagram on page 586.


about 5.6 ft

10.6

14.4

yes; 6.92 + 9.22 = 11.52

11-2

11-3

Distance and Midpoint FormulasDistance and Midpoint FormulasPRE-ALGEBRA LESSON 11-3PRE-ALGEBRA LESSON 11-3

Find the number halfway between 0.784 and 0.76.

0.772



Write the coordinates of each point.1. A 2. D 3. G 4. J


11-3


Solutions

1. A (–3, 4) 2. D (0, 3) 3. G (–4, –2) 4. J (3, –1)

11-3

Distance and Midpoint FormulasDistance and Midpoint Formulas

Find the distance between T(3, –2) and V(8, 3).


The distance between T and V is about 7.1 units.

Round to the nearest tenth.d 7.1

Find the exact distance.50d =

Simplify.d = 52 + 52

Replace (x2, y2) with (8, 3) and (x1, y1) with (3, –2).

d = (8 – 3)2 + (3 – (–2 ))2

Use the Distance Formula.d = (x2 – x1)2 + (y2 – y1)2

Quick Check

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)2WX =

1 + 9 = 10=

Replace (x2, y2) with (–2, –1) and (x1, y1) with (–3, 2).

Simplify.

(4 – (–2))2 + (0 – (–1)2XY =

36 + 1 == Simplify.37

Replace (x2, y2) with (4, 0) and (x1, y1) with (–2, –1).

11-3


(continued)


9 + 25 ==

(1 – 4)2 + (5 – 0)2YZ =

Simplify.

Replace (x2, y2) with (1, 5) and (x1, y1) with (4, 0).

34

(–3 – 1)2 + (2 – 5)2ZW =

Simplify.

Replace (x2, y2) with (–3, 2) and (x1, y1) with (1, 5).

= 16 + 9 = 25 = 5

11-3


(continued)


The perimeter is about 20.1 units.

perimeter = + + + 5 20.1343710

Quick Check

11-3


Find the midpoint of TV.


Use the Midpoint Formula.x1 + x2

2y1 + y2

2,

Replace (x1, y1) with (4, –3) and(x2, y2) with (9, 2).

= ,4 + 92

–3 + 22

Simplify the numerators.= ,132

–12

Write the fractions in simplest form.= 6 , –12

12

The coordinates of the midpoint of TV are 6 , – .12

12

Quick Check

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).


16

9.1; (–2 , – )12

12

13.6; (1 , 2)12

11-3

11-4

Problem Solving Strategy: Write a Proportion Problem Solving Strategy: Write a Proportion PRE-ALGEBRA LESSON 11-4PRE-ALGEBRA LESSON 11-4

Use these numbers to write as many proportions as you can: 5, 8, 15, 24

515

= 15 5

824

24 8

58

1524=, , 24

15= , 8

5=



Solve each proportion.

1. = 2. =

3. = 4. =

13

a12

h5

2025

14

8x

27

c35


11-4


Solutions

1. = 2. =

3 • a = 1 • 12 25 • h = 5 • 20 3a = 12 25h = 100

a = 4 h = 4

3. = 4. =

1 • x = 4 • 8 7 • c = 2 • 35 x = 32 7c = 70

c = 10

13

a12

h5

2025

14

8x

27

c35

11-4

Problem Solving Strategy: Write a Proportion Problem Solving Strategy: Write a Proportion

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

Problem Solving Strategy: Write a Proportion Problem Solving Strategy: Write a Proportion

(continued)


Write a proportion using the legs of the similar right triangles.

4x = 24(8) Write cross products.

4x = 192 Simplify.

x = 48 Divide each side by 4.

The height of the building is 48 ft.

= Write a proportion.8x

424

Quick Check

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?


48 cm

about 53 ft

20 ft

11-4

11-5

Special Right TrianglesSpecial Right TrianglesPRE-ALGEBRA LESSON 11-5PRE-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



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


11-5


Solutions

1. c2 = a2 + b2 2. a2 + b2 = c2

c2 = 62 + 82 92 + b2 = 152

c2 = 100 81 + b2 = 225 c = 100 = 10 m b2 = 144

b = 144 = 12 m

3. c2 = a2 + b2 4. a2 + b2 = c2

c2 = 272 + 362 482 + b2 = 602

c2 = 2025 2304 + b2 = 3600 c = 2025 = 45 m b2 = 1296

b = 1296 = 36 m

11-5

Special Right TrianglesSpecial Right Triangles

Find the length of the hypotenuse in the triangle.


hypotenuse = leg • 2 Use the 45°-45°-90° relationship.

y = 10 • 2 The length of the leg is 10.

The length of the hypotenuse is about 14.1 cm.

14.1 Use a calculator.

Quick Check

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 • 2 Use the 45°-45°-90° relationship.

y = 20 • 2 The length of the leg is 20.

The diagonal length is about 28.3 in.

28.3 Use a calculator.

Quick Check

11-5

Special Right TrianglesSpecial Right Triangles

Find the missing lengths in the triangle.


The length of the shorter leg is 7 ft. The length of the longer leg is about 12.1 ft.

hypotenuse = 2 • shorter leg14 = 2 • b The length of the hypotenuse is 14.

= Divide each side by 2.

7 = b Simplify.

142

2b2

longer leg = shorter leg • 3

a = 7 • 3 The length of the shorter leg is 7.

a 12.1 Use a calculator.

Quick Check

11-5

Find each missing length.

1. Find the length of the legs of a 45°-45°-90° triangle with a hypotenuse of 4 2 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

about 22.6 in.

3 3 in.

11-5

11-6

Sine, Cosine, and Tangent RatiosSine, Cosine, and Tangent RatiosPRE-ALGEBRA LESSON 11-6PRE-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.



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?


11-6


Solutions

1. = 2. =

6 • 20 = 8 • x 22 • 14 = 12 • x

120 = 8x 308 = 12x

= =

x = 15 ft x = 25 ft

68

x20

1222

14x

1208

8x8

30812

12x12

23

11-6

Sine, Cosine, and Tangent RatiosSine, Cosine, and Tangent Ratios

Find the sine, cosine, and tangent of A.


sin A = = = opposite

hypotenuse35

1220

cos A = = = adjacent

hypotenuse45

1620

tan A = = = oppositeadjacent

34

1216

Quick Check

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 pressthe key labeled SIN, COS, or TAN.

cos 18° 0.9511

tan 18° 0.3249

sin 18° 0.3090

Table: Find 18° in the first column. Lookacross to find the appropriate ratio.

Quick Check

11-6

Sine, Cosine, and Tangent RatiosSine, Cosine, and Tangent Ratios

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.

w(sin 40°) = 10 Multiply each side by w.

The hypotenuse is about 15.6 cm long.

w 15.6 Use a calculator.

sin A = Use the sine ratio.opposite

hypotenuse

sin 40° = Substitute 40° for the angle, 10 forthe height, and w for the hypotenuse.

10w

w = Divide each side by sin 40°.10

sin 40°

Quick Check

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.


about 10.5

about 18.3

1213

, ,513

125

sin 72° 0.9511; cos 72° 0.3090; tan 72° 3.0777

11-6

11-7

Angles of Elevation and DepressionAngles of Elevation and DepressionPRE-ALGEBRA LESSON 11-7PRE-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



Find each trigonometric ratio.

1. sin 45° 2. cos 32°

3. tan 18° 4. sin 68°

5. cos 88° 6. tan 84°


11-7


Solutions

1. sin 45° 0.7071 2. cos 32° 0.8480

3. tan 18° 0.3249 4. sin 68° 0.9272

5. cos 88° 0.0349 6. tan 84° 9.5144

11-7

Angles of Elevation and DepressionAngles of Elevation and Depression

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?


30(sin 52°) = h Multiply each side by 30.

The kite is about 24 yd from the ground.

Draw a picture.

24 h Simplify.

sin A = Choose an appropriate trigonometric ratio.

oppositehypotenuse

sin 52° = Substitute.h

30

Quick Check

11-7


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?

30(tan 61°) = h Multiply each side by 30.

54 + 6 = 60 Add 6 to account for the heightof Greg’s eyes from the ground.

The tree is about 60 ft tall.

Draw a picture.

54 h Use a calculator or a table.

Choose an appropriate trigonometric ratio.

oppositeadjacenttan A =

Substitute 61 for the angle measure and 30 for the adjacent side.

h30tan 61° =

Quick Check

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).

d • tan 3° = 1.5 Multiply each side by d.

tan 3° = Choose an appropriate trigonometric ratio.1.5d

11-7


(continued)


The airplane is about 28.6 mi from the airport.

= Divide each side by tan 3°.d • tan 3°tan 3°

1.5tan 3°

d = Simplify.1.5

tan 3°

d 28.6 Use a calculator.

Quick Check

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

square roots and irrational numbers

Documents

square roots

irrational numberssimplify

irrational numbersidentify

irrational numbers11

quick check11

distance d

pythagorean theorem11

h feet