’s
Right Triangles
In the geometric sequence 2, 10, 50, the middle term is called the Geometric Mean.
If
a x
 , and a, b, & x are positive numbers, then x is called the geometric mean between a & b.
x b
C
D
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 to each other.
A
B
(ie: ABC ~ ADB ~ BDC )
Corallary:
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.
CD BD

BD AD
Corallary:
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 the leg.
AC CB

CB CD
.
and
AC AB

AB AD
’s
B
e
c
d
a
A
Pythagorean Theorem:
In a right triangle, the square of the hypotenuse is
equal to the sum of the squares of the legs.
a2  b2  c2
C
b
Converse of Pythagorean 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.
Pythagorean Triplets:
3-4-5
5-12-13
8-15-17
7-24-25
9-40-41
11-60-61
Proving whether a triangle is right, acute, or obtuse using only knowledge of the triangle’s sides.
Theorem:
If a 2  b 2  c 2 , then mC  90 o and ABC is a right triangle.
Theorem:
If a 2  b 2  c 2 , then mC  90 o and ABC is acute.
Theorem:
If a 2  b 2  c 2 , then mC  90 o and ABC is obtuse.
Special Right Triangles:
45-45-90 Theorem:
45o
In a 45o - 45o - 90o triangle, the hypotenuse is
2 times as long as a leg.
a 2
a
45o
a
30-60-90 Theorem:
2a
30o
a 3
60o
a
In a 30o - 60o - 90o triangle, the hypotenuse is
twice as long as the shorter leg, and the longer
leg is 3 times as long as the shorter leg..
’s
Important Ratios in Right Triangles – TRIGONOMETRIC RATIOS
C
tan( A) 
BC
BC
AB
, sin( A) 
, cos(A) 
AB
AC
AC
Notes:
A
1) The above ratios are trigonometric functions. .
2) Tan, sin, & cos can be used to find missing sides in
right triangles.
B
SOH-CAH-TOA
tan 1 (
AB
CB
BC
)  C , sin 1 (
)  A , cos 1 (
)  C
BC
AC
AC
Notes:
3) The above equations are known as the inverse
trigonometric functions.
4) Tan-1, sin-1, & cos-1 can be used to find missing
angles in right triangles.
Practical method of approaching problems involving
trigonometric functions.
A) Decide what you are looking for (a side or an angle).
B) Pick the proper equation set (ex: TRIG or TRIG-1?)
C) Go to the angle in question and use it as a reference to
mark the opposite and adjacent sides (the hypotenuse
is always the longest side in right triangles).
D) Use SOH-CAH-TOA to select the specific trig.
Function to use.
Angle of elevation: The angle formed
between the horizontal and the “line-ofsight” (when looking upward).

Angle of depression: The angle formed
between the horizontal and the “line-ofsight” (when looking downward).

’s
More Trigonometric Functions
1
AC
 Cosecant  =
=
sin  CB


Secant =
C
1
AC
=
cos  AB
Cotangent =

A
B
1
AB
=
tan  CB
More relationships between sin, cos, & tan
 sin 2   cos 2   1

1  cot 2   csc 2 
 1  tan 2   sec 2 
sin 2 
 tan  
cos 2 
2
 cot 2  
cos 2 
sin 2 
 sin(  ) =
1
csc 
cos(  ) =
1
sec 
tan(  ) =
1
cot 
 csc(  ) =
1
sin 
sec(  ) =
1
cos 
cot(  ) =
1
tan 
 tan(  ) =
sin 
cos 
cot(  ) =
cos 
sin 
’s
Contents of Test
Multiple Choice 24 pts
Find value of x of length of a segment 45pts
Explain/discuss questions 9 pts
Trig Identities
6 pts
Other questions 15 pts
Study hard and know your trig, 30-60-90/45-45-90 rules, Pythagorean trips,
and KEEP THE ALGEBRA CLAEN.
GOOD LUCK !!!