Transcript Ch 11, Right Triangles

Square Roots and Irrational Numbers
Lesson 11-1
Additional Examples
Simplify each square root.
a.
144
144 = 12
b. –
–
81
81 = – 9
Pre-Algebra
Square Roots and Irrational Numbers
Lesson 11-1
Pre-Algebra
Additional Examples
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.
d=
1.5h
Use the formula.
d=
1.5(20)
Replace h with 20.
d=
30
Multiply.
25 <
25 = 5
30 <
36
Find perfect squares close to 30.
Find the square root of the
closest perfect square.
The lifeguard can see about 5 miles to the horizon.
Square Roots and Irrational Numbers
Lesson 11-1
Additional Examples
Pre-Algebra
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. 0.3333 . . .
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. 0.1234567 . . .
irrational, because it neither terminates nor repeats
The Pythagorean Theorem
Lesson 11-2
Pre-Algebra
Additional Examples
Find c, the length of the hypotenuse.
c2 = a2 + b2
Use the Pythagorean Theorem.
c2 = 282 + 212
Replace a with 28, and b with 21.
c2 = 1,225
Simplify.
c=
1,225 = 35
Find the positive square root of each side.
The length of the hypotenuse is 35 cm.
The Pythagorean Theorem
Lesson 11-2
Pre-Algebra
Additional Examples
Find the value of x in the triangle.
Round to the nearest tenth.
a2 + b2 = c2
Use the Pythagorean Theorem.
72 + x2 = 142
Replace a with 7, b with x, and c with 14.
49 + x2 = 196
Simplify.
x2 = 147
x=
147
Subtract 49 from each side.
Find the positive square root of each side.
The Pythagorean Theorem
Lesson 11-2
Pre-Algebra
Additional Examples
(continued)
Then use one of the two methods below to
approximate 147 .
Method 1: Use a calculator.
A calculator value for
x
12.1
147 is 12.124356.
Round to the nearest tenth.
Method 2: Use a table of square roots.
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.
x
12.1
Estimate the nearest tenth.
The value of x is about 12.1 in.
The Pythagorean Theorem
Lesson 11-2
Pre-Algebra
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?
c2 = a2 + b2
Use the Pythagorean Theorem.
c2 = 102 + 102
Replace a with 10 (half the span), and b with 10.
c2 = 100 + 100
Square 10.
c2 = 200
Add.
c =
c
200
14.1
Find the positive square root.
Round to the nearest tenth.
The rafter length should be about 14.1 ft.
The Pythagorean Theorem
Lesson 11-2
Pre-Algebra
Additional Examples
Is a triangle with sides 10 cm, 24 cm, and 26 cm
a right triangle?
a2 + b2 = c2
Write the equation for the Pythagorean Theorem.
102 + 242
262
Replace a and b with the shorter lengths
and c with the longest length.
100 + 576
676
Simplify.
676 = 676
The triangle is a right triangle.
Distance and Midpoint Formulas
Lesson 11-3
Pre-Algebra
Additional Examples
Find the distance between T(3, –2) and V(8, 3).
d=
(x2 – x1)2 + (y2 – y1)2
Use the Distance Formula.
d=
(8 – 3)2 + (3 – (–2 ))2
Replace (x2, y2) with (8, 3)
and (x1, y1) with (3, –2).
d=
52 + 52
Simplify.
d=
50
Find the exact distance.
d
7.1
Round to the nearest tenth.
The distance between T and V is about 7.1 units.
Distance and Midpoint Formulas
Lesson 11-3
Pre-Algebra
Additional Examples
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.
WX =
=
XY =
=
(–2 – (–3))2 + (–1 – 2)2
Replace (x2, y2) with (–2, –1)
and (x1, y1) with (–3, 2).
1+9=
Simplify.
10
(4 – (–2))2 + (0 – (–1)2
Replace (x2, y2) with (4, 0)
and (x1, y1) with (–2, –1).
36 + 1 =
Simplify.
37
Distance and Midpoint Formulas
Lesson 11-3
Pre-Algebra
Additional Examples
(continued)
YZ =
=
ZW =
=
(1 – 4)2 + (5 – 0)2
Replace (x2, y2) with (1, 5)
and (x1, y1) with (4, 0).
9 + 25 =
Simplify.
34
(–3 – 1)2 + (2 – 5)2
Replace (x2, y2) with (–3, 2)
and (x1, y1) with (1, 5).
16 + 9 =
Simplify.
25 = 5
Distance and Midpoint Formulas
Lesson 11-3
Pre-Algebra
Additional Examples
(continued)
perimeter =
10 +
37 +
34 + 5
The perimeter is about 20.1 units.
20.1
Distance and Midpoint Formulas
Lesson 11-3
Pre-Algebra
Additional Examples
Find the midpoint of TV.
x1 + x2 y1 + y2
,
2
2
Use the Midpoint Formula.
=
4 + 9 –3 + 2
,
2
2
Replace (x1, y1) with (4, –3) and
(x2, y2) with (9, 2).
=
13 –1
,
2 2
Simplify the numerators.
1
2
= 6 ,–
1
2
Write the fractions in simplest form.
1
2
The coordinates of the midpoint of TV are 6 , –
1
.
2
Special Right Triangles
Lesson 11-5
Pre-Algebra
Additional Examples
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.
14.1
Use a calculator.
The length of the hypotenuse is about 14.1 cm.
Special Right Triangles
Lesson 11-5
Pre-Algebra
Additional Examples
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.
28.3
Use a calculator.
The diagonal length is about 28.3 in.
Special Right Triangles
Lesson 11-5
Pre-Algebra
Additional Examples
Find the missing lengths in the triangle.
hypotenuse = 2 • shorter leg
14 = 2 • b
The length of the hypotenuse is 14.
14 = 2b
2
2
Divide each side by 2.
7=b
longer leg = shorter leg •
a=7• 3
a 12.1
Simplify.
3
The length of the shorter leg is 7.
Use a calculator.
The length of the shorter leg is 7 ft. The length of the
longer leg is about 12.1 ft.
Sine, Cosine, and Tangent Ratios
Lesson 11-6
Pre-Algebra
Additional Examples
Find the sine, cosine, and tangent of
opposite
12
3
adjacent
16
4
12
3
sin
A = hypotenuse = 20 = 5
cos
A = hypotenuse =
=
20
5
tan
A = adjacent
opposite
= 16 = 4
A.
Sine, Cosine, and Tangent Ratios
Lesson 11-6
Additional Examples
Pre-Algebra
Find the trigonometric ratios of 18° using a scientific
calculator or the table on page 779. Round to four decimal
places.
sin 18°
0.3090 Scientific calculator: Enter 18 and press
the key labeled SIN, COS, or TAN.
cos 18°
0.9511
tan 18°
Table: Find 18° in the first column. Look
0.3249 across to find the appropriate ratio.
Sine, Cosine, and Tangent Ratios
Lesson 11-6
Pre-Algebra
Additional Examples
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.
opposite
sin A = hypotenuse
10
sin 40° =w
w(sin 40°) = 10
10
Use the sine ratio.
Substitute 40° for the angle, 10 for
the height, and w for the hypotenuse.
Multiply each side by w.
w = sin 40°
Divide each side by sin 40°.
w
Use a calculator.
15.6
The hypotenuse is about 15.6 cm long.
Angles of Elevation and Depression
Lesson 11-7
Pre-Algebra
Additional Examples
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 =
opposite
hypotenuse
h
sin 52° =30
30(sin 52°) = h
24 h
The kite is about 24 yd from the ground.
Choose an appropriate
trigonometric ratio.
Substitute.
Multiply each side by 30.
Simplify.
Angles of Elevation and Depression
Lesson 11-7
Pre-Algebra
Additional Examples
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.
opposite
Choose an appropriate
trigonometric ratio.
h
Substitute 61 for the angle measure
and 30 for the adjacent side.
tan A = adjacent
tan 61° =30
30(tan 61°) = h
54 h
54 + 6 = 60
The tree is about 60 ft tall.
Multiply each side by 30.
Use a calculator or a table.
Add 6 to account for the height
of Greg’s eyes from the ground.
Angles of Elevation and Depression
Lesson 11-7
Pre-Algebra
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)?
Draw a picture
(not to scale).
tan 3° =1.5
d
d • tan 3° = 1.5
Choose an appropriate trigonometric ratio.
Multiply each side by d.
Angles of Elevation and Depression
Lesson 11-7
Pre-Algebra
Additional Examples
(continued)
1.5
d • tan 3°
= tan 3°
tan 3°
1.5
Divide each side by tan 3°.
d = tan 3°
Simplify.
d
Use a calculator.
28.6
The airplane is about 28.6 mi from the airport.