P1246990958JDCdZ

1
Right Triangle FUN
A presentation by Mr. Owsley
2
Table of Topics

• Geometric Mean
• Pythagorean Theorem
• Special Right Triangles
• Basic Trigonometry
• Angles of Elevation and Depression
• Law of Sines
• Law of Cosines
• Resources
• Contact
• External Links

Goodbye!
3
Geometric Mean

• Things to know
• The geometric mean of two numbers is the square
  root of their product.
• You can use the geometric mean to find the
  altitude of a right triangle

Geometric Mean Problem Altitude of a right
triangle problem Ready for something a little
more challenging?
4
Geometric Mean Problem

• Find the geometric mean between 10 and 30.

First set up the problem
After cross multiplying, we get the
equation 300 x2
Therefore,
5
Altitude of a right triangle problem

• Solve for x in the picture below if AD 5 and DC
  20

First, set up the problem
Therefore, x2 100 x 10
6
Ready for something a little more challenging?
is the geometric mean between x and
y. Find x if y
First set up the proportion for the geometric
mean
Now, we must rationalize the denominator
Now, cross multiply
Simplifying again will yield
Simplifying yields
Now, solve for x
7
Pythagorean Theorem

• Things to know
• If a2 b2 c2, then ?ABC is a right triangle
• If ?ABC is a right triangle, then a2 b2 c2

Applying the Pythagorean Theorem Pythagorean
Triples Ready for something more challenging?
8
Applying the Pythagorean Theorem

• Find x in the picture below

By the Pythagorean Theorem, a2 b2 c2
Therefore, 32 x2 62
Which simplifies to 9 x2 36
x2 27
x 5.2
9
Pythagorean Triples

• Verify if the following is a Pythagorean Triple

48 16 64 64 64
However, we have a problem. For a set of three
numbers to be a Pythagorean triple, all three
numbers must be natural numbers. Our first
number is clearly not. Therefore, our numbers do
not form a Pythagorean Triple.
10
Ready for something a little more challenging?
Initially, a 25-foot ladder rests against a
vertical wall such that the top of the ladder is
24 feet from the ground. Then, Nathan moves the
base of the ladder farther out from the wall so
that the top of the ladder slides down until
resting against the wall at a point 20 feet above
the ground. Given that the wall is perpendicular
to the ground, how far did Nathan move the base
of the ladder?
When the top of the ladder rests against the wall
and the other end is touching the ground, the
ladder, wall, and ground together form a right
triangle of which the ladder is the hypotenuse.
When the top of the ladder touches the wall 24
feet above the ground, this triangle has one leg
of 24 feet and hypotenuse of 25 feet. Therefore,
its other leg is insert square root equation
here 7 feet.
11
Special Right Triangles
Things to know

• 45-45-90
• 30-60-90
• Ready for something more challenging?

12
45-45-90

• Solve for x

By our relationships between 45-45-90, we can
set up the following equation
13
30-60-90

• Solve for x

By our relationships between 30-60-90, we can set
up the following equation
14
Ready for something a little more challenging?

• Find all the missing sides in the diagram

Because ?ABC is a 30-60-90 triangle and
is opposite the 30 angle, we have AC AB
6
and BC 2(AB) 12.
Because ?BDC is a 45-45-90 triangle, we have BD
BC 12 and CD BC
12
Because ?EDC is a 30-60-90 triangle and
.
is the longer leg, we have CD DE
Since CD 12
, we have
DE
We also have EC 2ED 8
Click the mouse to see the completed figure
above
Finally, ?ECF is a 45-45-90 triangle with
hypotenuse EC 8
Therefore, the legs each have length
15
Basic Trigonometry

• Need to Know

Finding trigonometric ratios Evaluating
Expressions Solving for an angle? Ready for
something more challenging?
16
Finding trigonometric ratios

• Find sin Y, cos Y, tan Y
• Likewise for Z

17
Evaluating Expressions

• Solve for x

First, set up your trigonometric identity
18
Solving for an angle

• What about solving for an angle?
• How do we go about doing that?

Suppose we want to know the value of x given that
sin x 30
What about cos x 60?
cos-1(cos x) cos-1(60)
You can use your calculator by pressing the sin-1
button on your calculator.
Therefore, sin-1 (sin x) sin-1(30)
Notice how sin 30 cos 60 ! This will always
be true sin (90 x) cos x
19
Ready for something more challenging?

• Find, in degrees, the smallest positive angle x
  such that
• sin 3x cos 7x.

Since sin x cos (90 - x), We have sin 3x cos
(90 - 3x)
Therefore, cos (90 3x) cos 7x.
Meaning, 90 3x 7x x 9
20
Angles of Elevation and Depression

• Need to Know
• Angle of elevation- The angle between the line of
  sight and the horizontal when an observer looks
  upward.
• Angle of depression- The angle between the line
  of sight when an observer looks downward, and the
  horizontal.

Finding an angle of elevation Finding an angle
of depression Ready for something more
challenging?
21
Finding an angle of elevation

• A store has a ramp near its front entrance. The
  ramp measures 12 feet, and has a height of 3
  feet. What is the angle of elevation?

First, draw a picture representing the situation.
22
Finding an angle of depression

• After flying at an altitude of 500 meters, a
  helicopter starts to descend when its ground
  distance from the landing pad is 11 kilometers.
  What is the angle of depression for this part of
  the flight?

First, draw a picture representing the situation.
Since our horizontal lines are parallel, the
angle of elevation is the same as the angle
opposite our 500 meter side.
23
Ready for something more challenging?

• Ulura or Ayers Rock is a sacred place for
  Aborigines of the western desert of Australia.
  Kwan-Yong uses a theodolite to measure the angle
  of elevation from the ground to the top of the
  rock to be 15.85. He walks half a kilometer
  closer and measures the angle of elevation to be
  25.6. How high is Ayers Rock to the nearest
  meter?

Tan 25.6
Tan 15.85
(BC) Tan 25.6 DC
(AC) Tan 25.6 DC
AC tan 15.85 BC tan 25.6
Change AC to meters .5km 500 m
Ayers Rock is about 348 meters high.
24
Law of Sines

• Need to know

Solving for a side Solving for an angle Ready
for something more challenging?
25
Solving for a Side

• Solve for b

26
Solving for an angle

• Solve for x

27
Ready for something more challenging?

• Use the law of sines to show that in ?ABC, A
  B if and only if BC AC.

If one of the angles is obtuse, the proof is
clear from the shown diagram, where AB BC. The first inequality can be seen from the
Pythagorean Theorem.
If both angles are acute, we have BC/AC sin A /
sin B 1, So BC AC.
Similarly, if BC AC, we can deduce sin A sin
B, from which A B follows
28
Law of Cosines

• Need to Know

Two sides and the Included Angle Three
Sides Ready for something more challenging?
29
Two sides and the included angle

• Solve for a

Simply plug your values into the law of cosines.
30
Three Sides

• Solve for x

Again, simply plug your values into the law of
cosines.
31
Ready for something more challenging?

• If the sides of a triangle are in the ratio 4 6
  8, then find the cosine of the smallest angle.

Let the sides be 4x, 6x, and 8x. Since the
smallest angle is opposite the smallest side, we
apply the law of cosines to find
16x2 36x2 64x2 2(6x)(8x)cos ?
16x2 36x2 64x2 -2(6x)(8x)cos ?
7/8 cos ?
32
Resources

• Boyd, Cindy. Geometry. 1st. New York Glencoe,
  2004.
• Rusczyk, Richard. Introduction to Geometry.
  Alpine, CA AoPS Incorporated, 2006.

33
Contact Information

• Mr. Owsley
• E-mail gowsley_at_rockhursths.edu
• Phone   [phone] ext. 287

34
Other Links

• www.artofproblemsolving.com
• www.mathworld.com