Transcript Chapter 8

CHAPTER 8
Right Triangles
SECTION 8-1
Similarity in Right
Triangles
Geometric Mean
If a, b, and x are
positive numbers and
a/x = x/b, then x is
called the geometric
mean between a and
b and x = √ab
THEOREM 8 -1
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
Corollary 1
When the altitude is
drawn to the hypotenuse
of a right triangle, the
length of the altitude is
the geometric mean
between the segments of
the hypotenuse.
Corollary 2
When the altitude is drawn
to the hypotenuse of a
right triangle, each leg is
the geometric mean
between the hypotenuse
and the segment of the
hypotenuse that is
adjacent to that leg.
SECTION 8-2
The Pythagorean
Theorem
THEOREM
In a right triangle, the
square of the
hypotenuse is equal
to the sum of the
squares of the legs.
SECTION 8-3
The Converse of the
Pythagorean Theorem
Triangle
o In
order to have a
triangle, the sum of
any two sides must
be greater than the
third side.
THEOREM
If the square of one side
of a triangle is equal to
the sum of the squares
of the other two sides,
then the triangle is a
right triangle.
Some Common Right
Triangle Lengths
o 3,4,5
5,12,13
o 6,8,10
10,24,26
o 9,12,15
8,15,17
o 12,16,20 7,24,25
o 15,20,25
THEOREM 8 - 3
2
2
2
If c = a + b , then
mC = 90°, and ∆ABC
is right
THEOREM 8 - 4
2
2
2
If c < a + b , then
mC < 90°, and ∆ABC
is acute
THEOREM 8 - 5
2
2
2
If c > a + b , then
mC > 90°, and ∆ABC
is obtuse
SECTION 8-4
Special Right Triangles
THEOREM 8 - 6
In a 45°- 45°- 90°
triangle, the
hypotenuse is √2
times as long as a leg.
THEOREM 8 - 7
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.
SECTION 8-5
The Tangent and
Cotangent Ratios
TRIGONOMETRIC
RATIOS
For any right triangle,
there are six
trigonometric ratios
of the lengths of the
sides of a triangle.
Tangent A
Ratio: length of side
opposite A to the
length of side
adjacent to A
Cotangent A
Ratio: length of side
adjacent A to the
length of side opposite
to A
= 1/(tan A)
Examples
Find tan M
N
13
5
P
12
M
SECTION 8-6
The Sine, Cosecant,
Cosine, and Secant Ratios
Sine A
Ratio: length of side
opposite A to
length of
hypotenuse.
Examples
Find sin M
N
13
5
P
12
M
Examples
Find sin N
N
13
5
P
12
M
Cosecant A
Ratio: length of
hypotenuse of A to
length of side opposite
A.
= 1/(sin A)
Cosine A
Ratio: Length of side
adjacent A to the
length of
hypotenuse.
Examples
Find cos M
N
13
5
P
12
M
Secant A
Ratio: length of
hypotenuse of A to
length of side adjacent
to A.
= 1/(cos A)
SECTION 8-7
Applications of Right
Triangle Trigonometry
Angle of Depression
The angle between
the horizontal and
the line of sight
Angle of Elevation
The angle above the
horizontal and the
line of sight
END