Download Similar Right Triangles

9.1
Similar Right Triangles
Goals p Solve problems involving similar right triangles formed by
the altitude drawn to the hypotenuse of a right triangle.
p Use a geometric mean to solve problems.
THEOREM 9.1
C
If the altitude is drawn to the hypotenuse of a
right triangle, then the two triangles formed
are similar to the original triangle and to
each other.
A
D
B
TCBD S TABC, TACD S TABC, and TCBD S TACD
Example 1
Finding the Height of a Ramp
Ramp Height A ramp has a cross
section that is a right triangle. The
diagram shows the approximate
dimensions of this cross section.
Find the height h of the ramp.
K
4.7 ft
J
11 ft
h
M
11.7 ft
L
Solution
By Theorem 9.1, T JKL S T KML .
Use similar triangles to write a proportion.
KM
JK
h
4.7
KL
JL
Corresponding side lengths are in proportion.
11
11.7
Substitute.
11.7 h 11 ( 4.7 )
h ≈ 4.4
Cross product property
Solve for h.
Answer The height of the ramp is about 4.4 feet.
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Chapter 9 • Geometry Notetaking Guide
187
Checkpoint Write similarity statements for the three triangles
in the diagram. Then find the given length. Round decimals to
the nearest tenth, if necessary.
1. Find AD.
2. Find NQ.
C
N
26
24
A
D 18
Q
10
B
Sample answer:
TABC S TCBD S TACD;
32
M
P
Sample answer:
TMNP S TQNM S TQMP;
3.8
GEOMETRIC MEAN THEOREMS
THEOREM 9.2
C
In a right triangle, the altitude from the right
angle to the hypotenuse divides the
hypotenuse into two segments.
A
The length of the altitude is the
geometric mean of the lengths of the
two segments.
D
BD
CD
B
CD
AD
THEOREM 9.3
In a right triangle, the altitude from the
right angle to the hypotenuse divides the
hypotenuse into two segments.
CB
AB
CB
DB
The length of each leg of the right triangle is
the geometric mean of the lengths of the
hypotenuse and the segment of the hypotenuse
that is adjacent to the leg.
AB
AC
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AC
AD
Chapter 9 • Geometry Notetaking Guide
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Using a Geometric Mean
Example 2
Find the value of each variable.
a.
b.
2
x
6
8
y
4
Solution
a. Apply Theorem 9.2.
8
b. Apply Theorem 9.3.
62
y
x
4
x
8
y
32 x 2
y
2
y
2
x
32
16 y 2
42
x
4 y
Checkpoint Find the value of the variable.
3.
4.
6
t
9
17
17
t
315
17
5.
z
13
5
310
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Chapter 9 • Geometry Notetaking Guide
189
9.2
The Pythagorean Theorem
Goals p Prove the Pythagorean Theorem.
p Use the Pythagorean Theorem to solve problems.
VOCABULARY
Pythagorean triple A Pythagorean triple is a set of three
positive integers a, b, and c that satisfy the equation
c2 a2 b2.
THEOREM 9.4: PYTHAGOREAN THEOREM
In a right triangle, the square of the length
of the hypotenuse is equal to the sum of the
squares of the lengths of the legs.
c
a
b
c2 a2 b2
Example 1
Finding the Length of a Hypotenuse
Find the length of the hypotenuse of the right
triangle. Tell whether the side lengths form a
Pythagorean triple.
Solution
(hypotenuse)2 (leg)2 (leg)2
8
15
Pythagorean Theorem
x2 8 2 15 2
Substitute.
x2 64 225
Multiply.
x2 289
Add.
x 17
x
Find the positive square root.
Answer The length of the hypotenuse is 17 . Because the
side lengths 8, 15, and 17 are integers , they form a
Pythagorean triple.
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Chapter 9 • Geometry Notetaking Guide
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Finding the Length of a Leg
Example 2
Find the length of the leg of the right triangle.
20
15
Solution
(hypotenuse)2 (leg)2 (leg)2
x
Pythagorean Theorem
20 2 x2 15 2
Substitute.
400 x2 225
Multiply.
175 x2
Subtract 225 from each side.
175 x
Find the positive square root.
25 p 7 x
Use product property.
57
x
Simplify the radical.
Answer The length of the leg is 57
.
Checkpoint Find the value of x. Simplify answers that are
radicals. Then tell whether the side lengths form a
Pythagorean triple.
1.
2.
8
24
x
x
25
12
; No
413
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
7; Yes
Chapter 9 • Geometry Notetaking Guide
191
Example 3
Finding the Area of a Triangle
Find the area of the triangle to the
nearest tenth of a square inch.
12 in.
Solution
You are given that the base of the
triangle is 18 inches, but you do not know the
height h.
12 in.
h
18 in.
Because the triangle is isosceles, it can be divided
into two congruent right angles with the given
dimensions. Use the Pythagorean Theorem to find
the value of h.
12 2 9 2 h2
Pythagorean Theorem
144 81 h2
Multiply.
63 h2
12 in.
h
9 in.
Subtract 81 from each side.
63 h
Find the positive square root.
37
h
Simplify the radical.
Now find the area of the original triangle.
1
A bh
2
Area of a triangle
1
)
( 18 )( 37
2
Substitute.
≈ 71.4
Use a calculator.
3
7 in.
____
18 in.
Answer The area of the triangle is about 71.4 square inches.
Checkpoint Find the area of the triangle. Round your answer
to the nearest tenth.
3.
4.
14 cm
9 cm
9 cm
39.6 cm2
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
10 ft
21 ft
92.3 ft2
Chapter 9 • Geometry Notetaking Guide
192
9.3
The Converse of the
Pythagorean Theorem
Goals p Use the Converse of the Pythagorean Theorem to solve
problems.
p Use side lengths to classify triangles by their angle
measures.
THEOREM 9.5: CONVERSE OF THE PYTHAGOREAN THEOREM
If the square of the length of the
longest side of a triangle is equal to
the sum of the squares of the lengths
of the other two sides, then the
triangle is a right triangle.
B
c
a
C
b
A
If c2 a2 b2, then TABC is a right triangle.
Example 1
Verifying Right Triangles
Tell whether the triangle at the right is a
right triangle.
8
6
Solution
Let c represent the length of the longest side
of the triangle. Check to see whether the side
lengths satisfy the equation c2 a2 b2.
( 815
)2 ( 86
)2 24 2
8
15
24
8 2 p ( 15
)2 8 2 p ( 6
)2 24 2
64 p 15 64 p 6 576
960 384 576
960 960
Answer The triangle is
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a right triangle.
Chapter 9 • Geometry Notetaking Guide
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Checkpoint Tell whether the triangle is a right triangle.
1.
2.
33
2
38
65
The triangle is a right
triangle.
10
16
56
The triangle is not a right
triangle.
THEOREM 9.6
If the square of the length of the longest
side of a triangle is less than the sum of
the squares of the lengths of the other
two sides, then the triangle is acute .
If c2 < a2 b2, then TABC is acute .
A
c
b
C
a
B
THEOREM 9.7
If the square of the length of the longest
side of a triangle is greater than the sum
of the squares of the lengths of the other
two sides, then the triangle is obtuse .
If c2 > a2 b2, then TABC is obtuse .
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
A
c
b
C
a
B
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Classifying Triangles
Example 2
Decide whether the set of numbers can represent the side lengths
of a triangle. If they can, classify the triangle as right, acute,
or obtuse.
a. 28, 40, 48
b. 5.7, 12.2, 13.9
Solution
Compare the square of the length of the longest side with the sum
of the squares of the lengths of the two shorter sides.
a.
c2 ? a2 b2
Compare c2 with a2 b2.
48 2 ? 282 40 2
Substitute.
2304
?
784 1600
Multiply.
2304
<
2384
c2 is less than a2 b2.
Answer Because c2 < a2 b2, the triangle is acute .
b.
c2 ? a2 b2
13.9 2 ?
Compare c2 with a2 b2.
5.7 2 12.22
Substitute.
193.21
?
32.49 148.84
Multiply.
193.21
>
181.33
c2 is greater than a2 b2.
Answer Because c2 > a2 b2, the triangle is obtuse .
Checkpoint Can the numbers represent the side lengths of a
triangle? If so, classify the triangle as right, acute, or obtuse.
3. 16, 30, 34
yes; right
4. 8, 13, 22
no
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5. 6, 9, 12
yes; obtuse
Chapter 9 • Geometry Notetaking Guide
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9.4
Special Right Triangles
Goals p Find the side lengths of special right triangles.
p Use special right triangles to solve real-life problems.
VOCABULARY
Special right triangles Special right triangles are right
triangles whose angle measures are 45-45-90
or 30-60-90.
THEOREM 9.8: 45-45-90 TRIANGLE THEOREM
In a 45-45-90 triangle, the hypotenuse
is 2
times as long as each leg.
x
Hypotenuse 2
p leg
45
2x
45
x
THEOREM 9.9: 30-60-90 TRIANGLE THEOREM
In a 30-60-90 triangle, the hypotenuse is
twice as long as the shorter leg, and the
longer leg is 3
times as long as the
shorter leg.
Hypotenuse 2 p shorter leg
Longer leg 3
p shorter leg
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x
60
2x
30
3x
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Example 1
Finding the Hypotenuse in a 45- 45- 90 Triangle
Find the value of x.
By the Triangle Sum Theorem, the measure
of the third angle is 45 . The triangle is a
45 - 45 -90 right triangle, so the length x
times the length
of the hypotenuse is 2
of a leg.
p leg
Hypotenuse 2
Example 2
x
45
7
7
45 - 45 -90 Triangle Theorem
x 2
p 7
Substitute.
x 72
Simplify.
Side Lengths in a 30- 60- 90 Triangle
Find the values of s and t.
9
30
Because the triangle is a 30-60-90
triangle, the longer leg is 3
times the
length of the shorter leg.
p shorter leg
Longer leg 3
9 3
ps
9
s
3
3
9
p s
3
3
33
s
t
s
60
30-60-90 Triangle Theorem
Substitute.
Divide each side by 3
.
Multiply numerator and
.
denominator by 3
Simplify.
The length of the hypotenuse is twice the length of the
shorter leg.
Hypotenuse 2 p shorter leg
30-60-90 Triangle Theorem
t 2 p 33
Substitute.
t 63
Simplify.
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Chapter 9 • Geometry Notetaking Guide
197
Checkpoint Find the values of the variables.
1.
2.
y
x 45
10
3
45
4
3.
a
60
m
n
60 15
b
30
x4
a 10
m 153
y 42
b 20
n 30
Example 3
Finding the Area of a Window
Construction The window is a square.
Find the area of the window.
36
2 in.
Solution
First find the side length s of the window
by dividing it into two 45 - 45 -90
triangles. The length of the hypotenuse is
inches. Use this length to find s.
362
362
2
ps
36 s
45 - 45 -90 Triangle Theorem
Divide each side by 2
.
Use s 36 to find the area of the window.
A s2
Area of a square
36 2
Substitute.
1296
Multiply.
Answer The area of the window is 1296 square inches.
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Chapter 9 • Geometry Notetaking Guide
198
9.5
Trigonometric Ratios
Goals p Find the sine, the cosine, and the tangent of an acute angle.
p Use trigonometric ratios to solve real-life problems.
VOCABULARY
Trigonometric ratio A trigonometric ratio is a ratio of the
lengths of two sides of a right triangle.
Sine A sine is a trigonometric ratio, abbreviated as sin.
Cosine A cosine is a trigonometric ratio, abbreviated as cos.
Tangent A tangent is a trigonometric ratio, abbreviated
as tan.
Angle of elevation An angle of elevation is the angle that
your line of sight makes with a horizontal line when you
stand and look up at a point in the distance.
TRIGONOMETRIC RATIOS
Let TABC be a right triangle. The sine, the cosine, and the
tangent of acute aA are defined as follows.
a
side opposite aA
sin A c
hypotenuse
side adjacent to aA
b
cos A c
hypotenuse
a
side opposite aA
tan A side adjacent to aA
b
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
hypotenuse
c
B
side
a opposite
aA
A
b
C
side adjacent to aA
Chapter 9 • Geometry Notetaking Guide
199
Example 1
Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent
of aP.
P
8
17
Solution
R
15
The length of the hypotenuse is 17 . The
length of the side opposite aP is 15 , and the length of the side
adjacent to aP is 8 .
Q
15
opposite
sin P ≈ 0.8824
17
hypotenuse
adjacent
8
cos P ≈ 0.4706
hypotenuse
17
opposite
15
tan P 1.875
adjacent
8
Example 2
Trigonometric Ratios for 60
Find the sine, the cosine, and the tangent of 60.
Solution
Begin by sketching a 30-60-90 triangle as
shown at the right. To make the calculations simple,
choose 1 as the length of the shorter leg. From the
30-60-90 Triangle Theorem, it follows that the
and the length of the
length of the longer leg is 3
hypotenuse is 2 . Label these lengths in the diagram.
__
2
__3
60
1
3
opp.
sin 60 ≈ 0.8660
2
hyp.
1
adj.
cos 60 0.5
2
hyp.
opp.
tan 60 3
≈ 1.7321
adj.
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Chapter 9 • Geometry Notetaking Guide
200
Checkpoint Use the diagram at the right to find the
trigonometric ratio.
1. sin X
X
4
0.8
5
12
Z
16
20
Y
2. cos X
3
0.6
5
3. tan Y
3
0.75
4
Example 3
Indirect Measurement
Flag Pole You are measuring the height of a flag pole.
You stand 19 feet from the base of the pole. You
measure the angle of elevation from a point on the
ground to the top of the pole to be 64. Estimate the
height of the pole.
h
64
19 ft
Solution
opposite
tan 64 adjacent
Write trigonometric ratio.
h
tan 64 19
Substitute.
19 tan 64 h
19 ( 2.0503 ) h
38.9557 ≈ h
Multiply each side by 19 .
Evaluate tan 64.
Simplify.
Answer The height of the flag pole is about 39 feet.
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Chapter 9 • Geometry Notetaking Guide
201
9.6
Solving Right Triangles
Goals p Solve a right triangle.
p Use right triangles to solve real-life problems.
VOCABULARY
Solve a right triangle To solve a right triangle means to
determine the measures of all three angles and the lengths
of all three sides.
Solving a Right Triangle
Example 1
Solve the right triangle. Round decimals to the
nearest tenth.
B
Solution
Use the Pythagorean Theorem to find the length of
the hypotenuse c.
(hypotenuse)2 (leg)2 (leg)2
Pythagorean
Theorem
c2 4 2 8 2
Substitute.
c2 80
Simplify.
c
A
8
4
C
c 45
Find the positive square root.
c ≈ 8.9
Use a calculator to approximate.
Use a calculator to find the measure of aB.
(
4
8
)
2nd
TAN
≈ 26.6 aA and aB are complementary. The sum of their measures is 90 .
maA maB 90 maA 26.6 90 maA 63.4 aA and aB are complementary.
Substitute for maB.
Subtract.
Answer The side lengths are 4 , 8 , and 8.9 . The angle measures
are 26.6 , 63.4 , and 90 .
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Chapter 9 • Geometry Notetaking Guide
202
Solving a Right Triangle
Example 2
Q
Solve the right triangle. Round decimals to
the nearest tenth.
30
Solution
Use trigonometric ratios to find the values
of p and q.
opp.
sin Q hyp.
q
sin 53 30
P
53
q
p
R
adj.
cos Q hyp.
cos 53 30 sin 53 q
30 cos 53 p
30 ( 0.7986 ) ≈ q
30 ( 0.6018 ) ≈ p
24.0 ≈ q
18.1 ≈ p
p
30
aP and aQ are complementary. The sum of their measures is 90 .
maP maQ 90 aP and aQ are complementary.
maP 53 90 Substitute for maQ.
maP 37 Subtract.
Answer The side lengths of the triangle are 18.1 , 24.0 , and
30 . The angle measures are 37 , 53 , and 90 .
Checkpoint Solve the right triangle. Round decimals to the
nearest tenth.
1.
2. D
C
80
a
e
A
89
31
B
F
Side lengths: 39, 80, 89
Angle measures: 26.0,
64.0, 90
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
f
8
E
Side lengths: 4.8, 8, 9.3
Angle measures: 31, 59,
90
Chapter 9 • Geometry Notetaking Guide
203
Example 3
Solving a Right Triangle
7
4 ft
Sports When a hockey player is 35 feet
from the goal line, he shoots the puck
35 ft
directly at the goal. The angle of elevation
at which the puck leaves the ice is 7. The
height of the goal is 4 feet. Will the player score a goal?
Solution
Begin by finding the height h of the puck at the goal line. Use a
trigonometric ratio.
opp.
tan 7 adj.
tan 7 h
Write trigonometric ratio.
Substitute.
35
35 tan 7 h
Multiply each side by 35 .
35 ( 0.1228 ) ≈ h
Use a calculator.
4.3 ≈ h
Multiply.
Answer Because the height of the puck at the goal line ( 4.3 feet)
is greater than the height of the goal (4 feet), the player will not
score a goal.
Checkpoint Complete the following exercise.
3. A hockey player is 27 feet from the goal line. He shoots the
puck directly at the goal. The height of the goal is 4 feet. What
is the maximum angle of elevation at which the player can
shoot the puck and still score a goal?
4 ft
x
27 ft
8.4
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Chapter 9 • Geometry Notetaking Guide
204
9.7
Vectors
Goals p Find the magnitude and direction of a vector.
p Add vectors.
VOCABULARY
Magnitude of a vector The magnitude of a vector is the
distance from the initial point to the terminal point.
Direction of a vector The direction of a vector is determined
by the angle that the vector makes with a horizontal line.
Equal vectors Two vectors are equal when they have the
same magnitude and direction.
Parallel vectors Two vectors are parallel when they have the
same or opposite direction.
Sum of vectors The sum of two vectors is a vector that joins
the initial point of the first vector and the terminal point of
the second vector.
Example 1
Finding the Magnitude of a Vector
P(4, 3) and Q(2, 1) are the initial and terminal points of a
vector. Draw PQ
&) in a coordinate plane. Then find its magnitude.
Solution
Component form x2 x1, y2 y1
PQ
&) 2 (4) , 1 3 y
P
3
6 , 4 Use the Distance Formula to find
the magnitude.
PQ
&) 1
3
1
x
Q
[ 2 ( 4 )] 2 ( 1 3 )2
52 ≈ 7.2
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Chapter 9 • Geometry Notetaking Guide
205
Describing the Direction of a Vector
Example 2
The vector CD
&) describes the velocity of
a moving hot air balloon. The scale on
each axis is in miles per hour.
D
N
y
a. Find the speed of the balloon.
5
b. Find the direction it is traveling
relative to east.
W
C
5
x
E
S
Solution
a. The magnitude of the vector CD
&) represents the balloon’s speed.
Use the Distance Formula.
CD
&) (30
5)2 (25 5)2 1025
≈ 32.0
Answer The speed of the balloon is about 32 miles per hour.
b. The tangent of the angle formed by
the vector and a line drawn parallel
20
to the x-axis is , or 0.8 . Use a
25
calculator to find the angle measure.
0.8
2nd
TAN
D
N
y
20
?
5
W
≈ 38.7 C
25
5
x
E
S
Answer The balloon is traveling in a
direction about 38.7 north of east.
Checkpoint Complete the following exercise.
1. The vector represents the velocity
of a moving hot air balloon. The
scale on each axis is in miles per
hour. Find the balloon’s speed and
direction relative to west.
25.5 miles per hour; 78.7 south
of west
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.
N
J
W
5 y
5 x
K
E
S
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Example 3
Identifying Equal and Parallel Vectors
In the diagram, these vectors have the
****) , CD
&*) , IJ
**) .
same direction: AB
****) , IJ
**) .
These vectors are equal: AB
y
F
B
D
G
A
1
These vectors are parallel:
C
1
1
****) , CD
&*) , GH
&*) , IJ
**) .
AB
E
H
1
3
x
J
I
ADDING VECTORS
Sum of Two Vectors
The sum of u
*) a1, b1 and *v) a2, b2 is
u
*) *v) a1 a2 , b1 b2 .
Example 4
Finding the Sum of Two Vectors
Let u
*) 6, 2 and *v) 8, 7. Find
the sum u
*) *v).
y
5
3
Solution
To find the sum, add the horizontal
components and add the vertical
components of u and v.
uv
1
1
v
3
5
x
u
u
*) *v) 6 (8) , 2 7 2 , 5 Checkpoint Find the sum of the vectors.
2. 6, 0, 1, 3
7, 3
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3. 5, 2, 7, 6
2, 4
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