Download Chapter 9

Larson Geometry Reference – Chapter 9
Definitions
Term/Concept
*Angle of Depression
*Angle of Elevation
*Geometric Mean
Description
The angle that a downward line of sight makes with a horizontal line.
The angle that an upward line of sight makes with a horizontal line.
The geometric mean of two positive numbers a and b is the positive number x that satisfies
the proportion
*Pythagorean Triple
Solving a Right Triangle
Sin(A) = opposite/hypotenuse
Some = old/horse
Caught = a/horse
Taking = oats/away
2
.
2
Consists of three positive integers a, b, c such that a + b = c . The name comes from the
Pythagorean theorem, which states that any right triangle with integer side lengths yields a
Pythagorean triple. The converse is also true: every Pythagorean triple determines a right
triangle with the given side lengths. (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)
To solve a right triangle you need to find all the unknown side lengths and the unknown
angle measures.
The ratio of the leg opposite angle A and the hypotenuse of the triangle. It is the measure
of the leg opposite to angle A divided by the length of the hypotenuse, assuming A is not a
right angle
B
BC
Opposite
Old


AB Hypotenuse Horse
C
A
The ratio of the leg adjacent to angle A and the hypotenuse of the triangle. It is the measure
of the leg adjacent to angle A divided by the length hypotenuse, assuming A is not a right
angle
B
cos( A) 
Tan(A) = opposite/adjacent
and
2
sin A 
Cos(A) = adjacent/hypotenuse
. Solving for x we have
AC
Adjacent
A


AB Hypotenuse Horse
C
A
The ratio of the leg opposite angle A and the leg adjacent to angle A. It is the measure of
the leg opposite to angle A divided by the length of the leg adjacent angle A, assuming A is
not a right angle.
B
tan( A) 
BC Opposite Oats


AC Adjacent Away
C
-1
A
Sin (A)=Inverse Sine=Arc Sine
The ratio of the hypotenuse and the leg opposite to angle A. It is the measure of the
hypotenuse divided by the length of the leg opposite to angle A. It is used to find the angle
when the lengths of the sides are known.
Cos-1(A)=Inverse Cosine=Arc Cosine
The ratio of the hypotenuse and the leg adjacent to angle A. It is the measure of the
hypotenuse divided by the length of the leg adjacent to angle A. It is used to find the angle
when the lengths of the sides are known.
Tan-1(A)=Inverse Tangent = Arc
Tangent
The ratio of the leg adjacent to angle A, and the leg opposite angle A. It is the measure of
the length of the leg opposite to angle A divided by the length of the leg adjacent to angle
A. It is used to find the angle when the lengths of the sides are know.
Trigonometric Ratio
A ratio of the lengths of two sides in a right triangle that are related to a trigonometric
function.
Larson Geometry Reference – Chapter 9
Formulas
Term/Concept
Description
The area of any triangle is given by one-half the product of the lengths of two sides times the
sine of their included angle.
Area of a Triangle
B
c
a
A
C
b
Given ∆ABC with a, b and c representing the measures of sides opposite angles with measures
Law of Cosines
A, B, and C, respectively. These equations are true:
a 2  b 2  c 2  2bc cos( A)
B
b 2  a 2  c 2  2ac cos( B)
c
c 2  a 2  b 2  2ab cos(C )
A
a
b
C
Given ∆ABC with a, b and c representing the measures of sides opposite angles with measures
Law of Sines
A, B, and C, respectively. The following equality is true:
sin( A) sin( B) sin( C )


a
b
c
Theorems
Theorems
Pythagorean Theorem
Pythagorean Theorem
Converse
*Pythagorean
Inequalities Theorem
*Right Triangle
Similarity
*Geometric Mean
Altitude Theorem
*Geometric Mean Leg
Theorem
*45-45-90 Triangle
Theorem
*30-60-90 Triangle
Theorem
In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the
hypotenuse.
If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the
longest side, then the triangle is a right triangle.
For any triangle ABC where c is the length of the longest side then: If
acute, if
then the triangle is
then the triangle is obtuse.
If an altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two
triangles formed are similar to the given triangle and to each other.
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 of the
hypotenuse.
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.
In a 45-45-90 triangle, the hypotenuse is
2 times as long as a leg.
In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is
as long as the shorter leg.
3 times