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Geometry 2
Unit 3: Right Triangles and
Trigonometry
Geometry 2
3.1 Similar Right Triangles
Similar Right Triangles

Altitude from Hypotenuse Theorem

If the altitude is drawn to the hypotenuse
of a right triangle, then the two triangles
formed are similar to the original triangle
and each other.
∆CDB ~ ∆ACB, ∆ACD ~ ∆ABC, and
∆CBD ~ ∆ACD
Similar Right Triangles
Example 1


A roof has a cross section that is a right triangle.
The diagram shows the approximate dimensions
of this cross section.

B

Identify the similar triangles in the diagram.
Find the height h of the roof.
12.3m
7.6 m
A
h
D
14.6 m
C
Similar Right Triangles

Geometric Mean

The geometric mean of two numbers a and
b is the positive number x such that
a
x

x
b
Similar Right Triangles

Geometric Mean Length of the Altitude
Theorem

In a right triangle, the altitude from the right angle to
the hypotenuse divides the hypotenuse into two
segments.
The length of the altitude is the geometric mean of the
lengths of the two segments.
BD = CD
CD
AD
Similar Right Triangles

Geometric Mean Length of Legs Theorem

In a right triangle, the altitude from the right angle to
the hypotenuse divides the hypotenuse into two
segments.
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 = CB
CB
DB
AB = AC
AC
AD
Similar Right Triangles

Example 2

Find the value of x.
x
6
10
Similar Right Triangles

Example 3

Find the value of y.
5
y
8
Similar Right Triangles

Example 4




To estimate the height of a
statue, your friend holds a
cardboard square at eye level.
She lines up the top edge of
the square with the top of the
statue and the bottom edge
with the bottom of the statue.
You measure the distance from
the ground to your friends eye
and the distance from your
friend to the statue.
In the diagram, XY = h – 5.1 is
the difference between the
statues height h and your
friends eye level. Solve for h.
X
h
W
Y
9.5 ft
5.1 ft
Z
Geometry 2
3.2 The Pythagorean Theorem
The Pythagorean Theorem

The Pythagorean Theorem


In a right triangle, the sum of the squares
of the legs equals the square of the
hypotenuse.
c2 = a2 + b2, where a and b are legs and c
is the hypotenuse.
c
a
b
The Pythagorean Theorem

Pythagorean Triple


When the sides of a right triangle are all
integers it is called a Pythagorean triple.
3,4,5 make up a Pythagorean triple since
52 = 32 + 42.
The Pythagorean Theorem

Example 1

Find the unknown side lengths. Determine
if the sides form a Pythagorean triple.
48
x
6
8
y
50
The Pythagorean Theorem

Example 2

Find the unknown side lengths. Determine
if the sides form a Pythagorean triple.
p
q
50
100
90
90
The Pythagorean Theorem

Example 3

Find the unknown side lengths. Determine
if the sides form a Pythagorean triple.
e
d
2
15
17
3
The Pythagorean Theorem

Example 4

Find the unknown side lengths. Determine if the
sides form a Pythagorean triple.
g
f
5 3
5
4 3
8
The Pythagorean Theorem

Example 5

Find the area of the triangle to the nearest
tenth of a meter.
8m
h
10 m
8m
The Pythagorean Theorem

Example 6


The two antennas shown in the diagram are
supported by cables 100 feet in length.
If the cables are attached to the antennas 50 feet
from the ground, how far apart are the antennas?
cable
50 ft
100 ft
100 ft
50 ft
Geometry 2
3.3 The Converse of the
Pythagorean Theorem

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.
c
a
b
If c2 = a2 + b2, then ∆ABC is
a right triangle.
a
a
b
If a and b stay the same
length and we make the
angle between them
smaller, what happens to
c?
b
a
a
b
If a and b stay the same
length and we make the
angle between them
bigger, what happens to
c?
b

Acute Triangle Theorem

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.
A
c
b
C
a
B
If c2 < a2 + b2, then ∆ABC is
acute.

Obtuse Triangle Theorem

If the square of the length of the longest
side of a triangle is more than the sum of
the squares of the lengths of the other two
sides, then the triangle is obtuse.
A
c
b
C
If c2 > a2 + b2, then ∆ABC is
acute.
a
B
Classifying Triangles





Let c be the biggest side of a triangle, and
a and b be the other two sides.
If c2 = a2 + b2, then the triangle is right.
If c2 < a2 + b2, then the triangle is acute.
If c2 > a2 + b2, then the triangle is obtuse.
*** If a + b is not greater than c, a
triangle cannot be formed.
Example 1

Example 1

Determine what type of triangle, if any, can
be made from the given side lengths.

7, 8, 12

11, 5, 9
Example 2

Example 2

Determine what type of triangle, if any, can
be made from the given side lengths.

5, 5, 5

1, 2, 3
Example 3

Example 3

Determine what type of triangle, if any, can
be made from the given side lengths.

16, 34, 30

9, 12, 15
Example 4

Example 4

Determine what type of triangle, if any, can
be made from the given side lengths.

13, 5, 7

13, 18, 22
Example 5

Example 5

Determine what type of triangle, if any, can be
made from the given side lengths.
 4, 8, 4 3

5, 5 2 , 5

Example 6

You want to make sure a wall of a room is
rectangular.


A friend measures the four sides to be 9 feet, 9 feet, 40
feet, and 40 feet. He says these measurements prove
the wall is rectangular. Is he correct?
You measure one of the diagonals to be 41 feet. Explain
how you can use this measurement to tell whether the
wall is rectangular.
Geometry 2
3.4 Special Right Triangles
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
3
2
2
3
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
5
4
4
5
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
7
6
6
7
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
½
300
300
½
Special Right Triangles

Use the pattern you noticed on the
previous page to find the length of the
hypotenuse in terms of x.
x
x
Special Right Triangles

45º-45º-90º Triangles Theorem

In a 45º-45º-90º triangle, the hypotenuse is
times each leg.
x 2
x
x
2
Special Right Triangles

Solve for each missing length. What
pattern, if any do you notice?
10
10
10
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
8
8
8
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
6
6
6
Special Right Triangles

Solve for each missing side. What
pattern, if any do you notice?
50
50
50
Special Right Triangles
2x
2x
2x
Special Right Triangles

30º-60º-90º Triangle Theorem

In a 30º-60º-90º triangle, the hypotenuse is twice
as long as the shortest leg, and the longer leg is
3 times as long as the shorter leg.
30º
2x
x 3
60º
x
Special Right Triangles

Example 1

Find each missing side length.
6
45º
45º
15
Special Right Triangles

Example 2
18
12
30º
45º
Special Right Triangles

Example 3
30º
44
12
30º
Special Right Triangles

Example 4


A ramp is used to unload trucks. How high
is the end of a 50 foot ramp when it is
tipped by a 30° angle?
By a 45° angle?
Special Right Triangles

Example 5

The roof on a doghouse is shaped like an
equilateral triangle with height 3 feet.
Estimate the area of the cross-section of
the roof.
Geometry 2
3.5 Trigonometric Ratios
Trigonometric Ratios



Name the side opposite angle A.
Name the side adjacent to angle A.
Name the hypotenuse.
A
C
B
Trigonometric Ratios

The 3 basic trig functions and their
abbreviations are



sine = sin
cosine = cos
tangent = tan
Trigonometric Ratios



sin =
opposite side
hypotenuse
SOH
cos = adjacent side
hypotenuse
CAH
tan =
TOA
opposite side
adjacent side
Trigonometric Ratios

Example 1

Find each trigonometric ratio.






sin A
cos A
tan A
sin B
cos B
tan B
A
5
3
C
4
B
Trigonometric Ratios

Example 2

Find the sine, the cosine, and the tangent of the
acute angles of the triangle. Express each value as
a decimal rounded to four decimal places.
D
25
7
E
24
F
Trigonometric Ratios

Example 3
Find the sine, cosine, and the tangent of
A.

B
18√2
18
C
18
A
Trigonometric Ratios

Example 4

Find the sine, cosine, and tangent of A.
B
10
5
C
5√3
A
Trigonometric Ratios

Example 5

Use the table of trig values to approximate
the sine, cosine, and tangent of 82°.
Trigonometric Ratios

Angle of Elevation

When you stand and look up at a point in
the distance, the angle that your line of
sight makes with a line drawn horizontally
is called the angle of elevation.
angle of depression
angle of elevation
Trigonometric Ratios

Example 6

You are measuring the height of a building.
You stand 100 feet from the base of the
building. You measure the angle of
elevation from a point on the ground to the
top of the building to be 48°. Estimate the
height of the building.
Trigonometric Ratios

Example 7

A driveway rises 12 feet over a distance d
at an angle of 3.5°. Estimate the length of
the driveway.
Geometry 2
3.6 Solving Right Triangles
Solving Right Triangles

Solving a Right Triangle


To solve a right triangle means to
determine the measures of all six parts.
You can solve a right triangle if you know:


Two side lengths
One side length and one acute angle measure
Solving Right Triangles

Example 1

Find the value of each variable. Round
decimals to the nearest tenth.
c
8
25º
b
Solving Right Triangles

Example 2

Find the value of each variable. Round
decimals to the nearest tenth.
c
b
42º
40
Solving Right Triangles

Example 3

Find the value of each variable. Round
decimals to the nearest tenth.
b
8
20º
a
Solving Right Triangles

Example 4

Find the value of each variable. Round
decimals to the nearest tenth.
c
b
17º
10
Solving Right Triangles

Example 5

During a flight, a hot air
balloon is observed by two
persons standing at points
A and B as illustrated in the
diagram. The angle of
elevation of point A is 28°.
Point A is 1.8 miles from
the balloon as measured
along the ground.


What is the height h of the
balloon?
Point B is 2.8 miles from
point A. Find the angle of
elevation of point B.
h
B
A