Right Triangles

1
Right Triangles

• Unit 8

2
Geometric mean
Special right triangles
Right triangles
Law of sines and cosines
Ratios in right triangles
Angles of elevation depression
3
Euclid

• 325 B.C. 265 B.C.
• Greece
• Wrote The Elements
• Geometry today

4
Pythagoras

• 569 B.C. 475 B.C.
• Greece
• First pure mathematician
• Secret society
• Pythagorean theorem

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The Right Triangle
hypotenuse
leg
leg
Right angle
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Theorem 8.1

• If an altitude is drawn from a right angle to
  the hypotenuse of a triangle, then the 2
  triangles formed are similar to the 1st triangle
  and to each other

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Theorem 8.1
C
?ABC ? ?DBA ?ABC ? ?DAC ?DBA ? ?DAC
D
A
B
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Geometric Mean

• The positive number x such that
• a x
• x b

9
Find the geometric mean between 6 and 12

• 6 x
• x 12
• x2 612
• x ?72 ? 8.49

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Theorem 8.2

• The measure an altitude drawn from a right
  angle to the hypotenuse of a triangle is the
  geometric mean between the measures of the 2
  segments of the hypotenuse

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Theorem 8.2
C
DC AD AD DB
D
A
B
12
Theorem 8.3

• The measure of a leg (of a triangle with an
  altitude drawn from the right angle to the
  hypotenuse of a triangle) is the geometric mean
  between the measures of the hypotenuse and the
  segment of the hypotenuse closest to that leg

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Theorem 8.3
C
CB AC AC CD
D
A
B
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Pythagorean Theorem

• One of the most famous theorems in mathematics.
• Relationship has been known for thousands of
  years.

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Pythagorean Theorem

• leg2 leg2 hypotenuse2
• a2 b2 c2
• c
• a
• b

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THE 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 2 a 2 b 2
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Pythagorean Theorem

• Find the legs and hypotenuse
• Square the legs (this is a and b)
• Add them together
• Square root them the length of the hypotenuse
  (this is c)

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1. Square both legs
1
2
3
4
5
6
7
8
4 ft
11
9
10
12
13
14
16
15
3 ft
4ft
1
3
2
4
5
6
3 ft
7
8
9
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2. Count the total squares
1
2
3
4
5
6
7
8
4 ft
11
9
10
12
13
14
16
15
3 ft
4ft
1
3
2
9 16 25
4
5
6
3 ft
7
8
9
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3. Put that number of squares on the hypotenuse
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4. Count the number of squares that touch the
hypotenuse.
5
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5.That number is the length of the hypotenuse.
5
Length 5
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Find the length of the hypotenuse of the right
triangle.
(hypotenuse)2 (leg)2 (leg)2
Pythagorean Theorem
x 2 5 2 12 2
Substitute.
x 2 25 144
Multiply.
x 2 169
Add.
x 13
Find the positive square root.
Because the side lengths 5, 12, and 13 are
integers, they form a Pythagorean triple.
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(hypotenuse)2 (leg)2 (leg)2
Pythagorean Theorem
14 2 7 2 x 2
Substitute.
196 49 x 2
Multiply.
147 x 2
Subtract 49 from each side.
Find the positive square root.
Use product property.
Simplify the radical.
25
Indirect Measurement
SUPPORT BEAM These skyscrapers are connected by
a skywalk with support beams. You can use the
Pythagorean Theorem to find the approximate
length of each support beam.
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Indirect Measurement
Each support beam forms the hypotenuse of a right
triangle. The right triangles are congruent, so
the support beams are the same length.
x 2 (23.26)2 (47.57)2
Pythagorean Theorem
Find the positive square root.
x ? 52.95
Use a calculator to approximate.
The length of each support beam is about
52.95 meters.
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piece of hyp. altitude altitude piece
of hyp.
piece of hyp. __leg___ leg
total hyp.
leg2 leg2 hypotenuse2 a2 b2 c2
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Special right triangles

• 45 45 90 Triangle
• 30 60 90 Triangle

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Consider a square with sides X.
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If we draw in diagonal well obtain two
triangles.
TOPICS
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Its a Special Right Triangle!
TOPICS
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Obtain the length of our diagonal
TOPICS
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This is fundamental yet powerful result.
TOPICS
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Example 1
becomes
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Example 2
becomes
36
Example 3
becomes
Why?
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Using the 45- 45- 90 relationship. . .
TOPICS
38
TOPICS
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Consider an equilateral triangle.
Equilateral triangles 1. Interior angles
each measure 60 2. All three sides have
equal length.
40
Bisect angle ACB by drawing a line segment from
vertex C to point D on side
TOPICS
41
Sides of the triangle 2X.
TOPICS
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30-60-90 triangle!
What is the length of the legs ?
TOPICS
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Use the Pythagorean Theorem
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Example 1
45
Example 2
46
15?3/2
15
15/2
TOPICS
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Example 3
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Using,
TOPICS
49
Special Right Triangles

• For 30-60-90 degree triangles.....
• The short leg is ALWAYS the side across from the
  30-degree angle
• Hypotenuse 2 short leg
• Long Leg ?(3) short leg
• For 45-45-90 degree triangles.....
• The triangle is isosceles.
• The two legs are congruent.
• Leg Leg and Hypotenuse ?(2) leg

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Greek word meaning
Triangle Measure
TRIGONOMETRY
51

• Ancient Greeks used trigonometry to measure the
  distance to the stars
• In class only right triangle trig!

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In 140 B.C. Hipparchus began to use and write
trigonometry
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The Journey of SohCahToa
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The Damaged Tipis
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?
57
SohCahToa
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X
S o h
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X
C a h
60
X
T o a
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Theyre Fixed !!
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(No Transcript)
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Hypotenuse
Leg opposite B
Leg adjacent to B
B
Sine of B length of leg opposite B

length of hypotenuse
64
5
3
B
4
Sin (B) leg opposite B
hypotenuse 3/5
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Hypotenuse
Leg opposite B
Leg adjacent to B
B
Cosine of B length of leg adjacent to B
length of hypotenuse
66
5
3
B
4
Cos (B) adjacent leg B
hypotenuse 4/5
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Hypotenuse
Leg opposite B
Leg adjacent to B
B
Tangent of B length of leg opposite B
length of leg adjacent to B
68
5
3
B
4
Tan (B) opposite leg
adjacent leg 3/4
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Hypotenuse
Leg opposite B
Leg adjacent to B
B
Sine of B length of leg opposite B

length of hypotenuse
Cosine of B length of leg adjacent to B
length of hypotenuse
Tangent of B length of leg opposite B
length of leg adjacent to B
70
Sample right triangle problems
1.)
x
20
A
B
60
z
30
y
Ø
C
Find the values to the nearest tenth of
B/A
A.) sin Ø _______ B.) cos Ø _______ C.) tan Ø
_______
A.) XY ________ B.) YZ ________
11.5
C/A
23.1
B/C
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APPLICATIONS
To avoid a steep descent, a plane flying at
30,000 ft. starts its descent 130 miles away from
the airport. For the angle of descent Ø, to be
constant, at what angle should the plane descend?
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tan Ø 30,000
5,280130
Ø
30,000 ft.
Ø
130 Miles
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An observer 5.2 km from a launch pad observes a
rocket ascending.
A. At a particular time the angle of elevation
is 37 degrees. How high is the rocket?
B. How far is the observer from the rocket?
C. What will the angle of elevation be when the
rocket reaches 30 km?
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At a particular time the angle of elevation is 37
degrees. How high is the rocket?
B
A
37
5.2
Tan 37 _A_
5.2
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How far is the observer from the rocket?
B
A
37
5.2
Cos 37 5.2
B
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What will the angle of elevation be when the
rocket reaches 30 km?
B
A
37
5.2
C. Tan 30
Ø
5.2
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A ship sails 340 kilometers on a bearing of 75
degrees.
A. How far north of its original position is the
ship? B. How far east of its original position
is the ship?
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B
A
A. Cos 75
340
A
340
B. Sin 75 B
340
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