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Name ___________
Geometry 2
Unit3: Right Triangles
and Trigonometry
Geometry 2
3.1 Similar Right Triangles
Unit3: Right Triangles and Trigonometry
C
Altitude from
Hypotenuse
Theorem
Example 1
A
D
B
∆CDB ~ ∆ACB, ∆ACD ~ ∆ABC, and
∆CBD ~ ∆ACD
A roof has a cross section that is a right triangle.
The diagram shows the approximate dimensions
of this cross section.
Identify the similar triangles in the diagram.
Find the height h of the roof.
B
12.3m
7.6 m
A
h
D
14.6 m
C
Geometric Mean
2
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
3
BD = CD
CD
AD
C
Geometric Mean
Length of the
Altitude
Theorem
Geometric Mean
Length of Legs
Theorem
A
D
B
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.
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.
Example 2
x
6
Example 3
10
5
y
8
4
Example 5
Find the area of the triangle to the nearest tenth of a meter.
8m
8m
h
10 m
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
5
Geometry 2
Unit3: Right Triangles and Trigonometry
3.2 The Pythagorean Theorem
The
In a right triangle, the sum of the legs squared equals
c
a
Pythagorean the hypotenuse squared.
Theorem
c2 = a2 + b2, where a and b are legs and c is the
b
hypotenuse.
Pythagorean When the sides of a right triangle are all integers it is
Triple
called a Pythagorean theorem.
3,4,5 make up a Pythagorean triple since
52 = 32 + 42.
Example 1
48
y
x
6
50
8
Example 2
q
p
90
50
100
90
Example 3
e
d
2
15
17
3
Example 4
g
f
5 3
4 3
8
5
6
7
8
Geometry 2
Unit3: Right Triangles and Trigonometry
3.3 The Converse of the Pythagorean Theorem
Converse of
If the square of the length of the longest side of a
c
a
the
triangle is equal to the sum of the squares of the
Pythagorean lengths of the other two sides, then the triangle is
b
Theorem
a right triangle.
If c2 = a2 + b2, then
∆ABC is a right
triangle.
Acute
Triangle
Theorem
Obtuse
Triangle
Theorem
Classifying
Triangles
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 ∆ABC is __________.
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.
If c2 > a2 + b2, then ∆ABC is __________.
A
c
b
C
B
a
A
c
b
C
a
B
Let c be the biggest side of a triangle, and a and b
be the other two side.
If c2 ____ a2 + b2, then the triangle is _________.
If c2 ____ a2 + b2, then the triangle is _________.
If c2 ____ a2 + b2, then the triangle is _________.
*** If a + b is not greater than c, a triangle cannot
be formed.
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
Determine what type of triangle, if any, can be
made from the given side lengths.
5, 5, 5
1, 2, 3
9
10
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
Determine what type of triangle, if any, can be
made from the given side lengths.
13, 5, 7
13, 18, 22
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.
11
Geometry 2
3.4 Special Right Triangles
Unit3: Right Triangles and Trigonometry
Solve for each missing side. What pattern, if any do you notice?
3
2
3
2
5
4
5
4
7
6
7
6
½
300
300
½
12
x
x
45º-45º-90º
Triangles
Theorem
In a 45º-45º-90º triangle, the hypotenuse is
times each leg.
x 2
x
x
Solve for each missing length. What pattern, if any do you notice?
10
10
10
8
6
6
6
8
8
50
50
50
13
2x
2x
2x
30º-60º-90º
Triangle
Theorem
In a 30º-60º-90º triangle, the hypotenuse is
____________ as long as the shortest leg, and
the longer leg is ____________ times as long as
the shorter leg.
30º
2x
x 3
60º
x
Example 1
Find each missing side length.
6
45º
45º
15
Example 2
18
12
30º
45º
14
Example 3
30º
44
12
30º
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?
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.
15
16
17
18
Geometry 2
3.5 Trigonometric Ratios
 Name the side opposite angle A.
 Name the side adjacent to angle A.
 Name the hypotenuse.
Unit3: Right Triangles and Trigonometry
A
B
C
 The 3 basic trig functions and their abbreviations are
 sine = sin
 cosine = cos
 tangent = tan
 sin =
opposite side
hypotenuse
 cos = adjacent side
hypotenuse
 tan =
opposite side
adjacent side
 Example 1
 Find each trigonometric ratio.
 sin A
A
 cos A
 tan A
5
3
 sin B
 cos B
 tan B
C
4
B
19
 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
F
24
E
 Example 3
 Find the sine, cosine, and the tangent of A.
B
18√2
18
C
A
18
 Example 4
 Find the sine, cosine, and tangent of A.
B
10
5
C
5√3
A
 Example 5
 Use the table of trig values to approximate the sine, cosine, and tangent of
82°.
20
 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.
depression
 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.
 Example 7
A driveway rises 12 feet over a distance d at an angle of 3.5°. Estimate the
length of the driveway.
21
22
Geometry 2
3.6 Solving Right Triangles
Unit3: Right Triangles and Trigonometry
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
 Example 1
 Find the value of each variable. Round decimals to the nearest tenth.
c
8
25
º
b
 Example 2
 Find the value of each variable. Round decimals to the nearest tenth.
c
b
42º
40
 Example 3
 Find the value of each variable. Round decimals to the nearest tenth.
b
8
20º
a
23
 Example 4
 Find the value of each variable. Round decimals to the nearest tenth.
c
b
17º
10
 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
24
25
26
27