pre-algebra b. – 81 square roots and irrational numbers simplify each square root. lesson 11-1 a....

Pre-AlgebraPre-Algebra

b. – 81

Square Roots and Irrational NumbersSquare Roots and Irrational Numbers

Simplify each square root.

Lesson 11-1

a. 144

144 = 12

– 81 = – 9

Additional Examples



Lesson 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< <

Additional Examples


c. 3

a. 49


Identify each number as rational or irrational. Explain.

Lesson 11-1

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

Additional Examples


The Pythagorean TheoremThe Pythagorean Theorem

Find c, the length of the hypotenuse.

Lesson 11-2

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

Additional Examples



Find the value of x in the triangle.

Round to the nearest tenth.

Lesson 11-2

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.

Additional Examples


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


(continued)

Lesson 11-2

The value of x is about 12.1 in.

Estimate the nearest tenth.x 12.1

Use the table on page 800. 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

Additional Examples



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?

Lesson 11-2

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

Additional Examples



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

a right triangle?

Lesson 11-2

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 for the Pythagorean Theorem.

676 = 676

Additional Examples


Distance and Midpoint FormulasDistance and Midpoint Formulas

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

Lesson 11-3

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

Additional Examples



Find the perimeter of WXYZ.

Lesson 11-3

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).

Additional Examples



(continued)

Lesson 11-3

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

Additional Examples



(continued)

Lesson 11-3

The perimeter is about 20.1 units.

perimeter = + + + 5 20.1343710

Additional Examples



Find the midpoint of TV.

Lesson 11-3

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

Additional Examples


Special Right TrianglesSpecial Right Triangles

Find the length of the hypotenuse in the triangle.

Lesson 11-5

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.

Additional Examples



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 • 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.

Additional Examples



Find the missing lengths in the triangle.

Lesson 11-5

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.

Additional Examples


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

Find the sine, cosine, and tangent of A.

Lesson 11-6

sin A = = = opposite

hypotenuse35

1220

cos A = = = adjacent

hypotenuse45

1620

tan A = = = oppositeadjacent

34

1216

Additional Examples



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 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.

Additional Examples



The diagram shows a doorstop in the shape of a wedge. What is the length of the hypotenuse of the doorstop?

Lesson 11-6

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°

Additional Examples


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?

Lesson 11-7

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

Additional Examples



Lesson 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° =

Additional Examples



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)?

Lesson 11-7

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

Additional Examples



(continued)

Lesson 11-7

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.

Additional Examples

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