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
Pre-AlgebraPre-Algebra
Square Roots and Irrational NumbersSquare Roots and Irrational Numbers
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
Pre-AlgebraPre-Algebra
c. 3
a. 49
Square Roots and Irrational NumbersSquare Roots and Irrational Numbers
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
Pre-AlgebraPre-Algebra
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
Pre-AlgebraPre-Algebra
The Pythagorean TheoremThe Pythagorean Theorem
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
Pre-AlgebraPre-Algebra
Then use one of the two methods below to approximate .147
The Pythagorean TheoremThe Pythagorean Theorem
(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
Pre-AlgebraPre-Algebra
The Pythagorean TheoremThe Pythagorean Theorem
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
Pre-AlgebraPre-Algebra
The Pythagorean TheoremThe Pythagorean Theorem
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
Pre-AlgebraPre-Algebra
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
Pre-AlgebraPre-Algebra
Distance and Midpoint FormulasDistance and Midpoint Formulas
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
Pre-AlgebraPre-Algebra
Distance and Midpoint FormulasDistance and Midpoint Formulas
(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
Pre-AlgebraPre-Algebra
Distance and Midpoint FormulasDistance and Midpoint Formulas
(continued)
Lesson 11-3
The perimeter is about 20.1 units.
perimeter = + + + 5 20.1343710
Additional Examples
Pre-AlgebraPre-Algebra
Distance and Midpoint FormulasDistance and Midpoint Formulas
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
Pre-AlgebraPre-Algebra
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
Pre-AlgebraPre-Algebra
Special Right TrianglesSpecial Right Triangles
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
Pre-AlgebraPre-Algebra
Special Right TrianglesSpecial Right Triangles
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
Pre-AlgebraPre-Algebra
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
Pre-AlgebraPre-Algebra
Sine, Cosine, and Tangent RatiosSine, Cosine, and Tangent Ratios
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
Pre-AlgebraPre-Algebra
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?
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
Pre-AlgebraPre-Algebra
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
Pre-AlgebraPre-Algebra
Angles of Elevation and DepressionAngles of Elevation and Depression
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
Pre-AlgebraPre-Algebra
Angles of Elevation and DepressionAngles of Elevation and Depression
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
Pre-AlgebraPre-Algebra
Angles of Elevation and DepressionAngles of Elevation and Depression
(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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