The right triangle has been used in trades for thousands of years. Ancient Egyptians found that they could always get a square corner using the 3-4-5 right triangle. (Carpenters still use the 3-4-5 triangle to square corners.) Later, a Greek mathematician named Pythagoras developed a formula to find the side lengths for any right triangles.
Pythagoras treated each side of a right triangle as if it was a side of a square. He found that the total area of the two smaller squares was equal to the area of the largest square for every right triangle.

Chapter 9
Right Triangles and Trigonometry

Mr. Daniel L. Mollette
Math Teacher
Duluth High School
Duluth, GA 30096

Email: daniel_mollette@gwinnett.k12.ga.us

Euclidean Geometry

Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.

Right Triangles and Trigonometry

Practice Quiz

What is the length of the hypotenuse of a right triangle given sides of 9 and 40?

What is the length of the hypotenuse of the
30-60-90 degree shown triangle below?

41

1681

31

15

30

7.5

What is the length of the hypotenuse of a square, if the side measures 3?

What is the length of the hypotenuse (c) problem #2 , if the m < A = 20º?

3

3Å3

3Å2

30

43.9

5.1

http://www.teacherlink.org/content/math/interactive/geoweb/lessons/sprttri/righttriangles.pdf

SOH - CAH - TOA
It is pronounced "so - ka - toe - ah".
The SOH stands for
"Sine of an angle is Opposite over Hypotenuse."
The CAH stands for
"Cosine of an angle is Adjacent over
Hypotenuse."
The TOA stands for
"Tangent of an angle is Opposite over Adjacent."

45-45-90 Triangle
Theorem The length of the
hypotenuse of a 45-45-90 triangle is Å2 times the length of a leg.

30-60-90 Triangles
A 30-60-90 triangle is another example of a special right triangle that has a 30 degree angle
and a 60 degree angle. "The hypotenuse and the longer leg in a 30-60-90 triangle can be found when the shorter leg is known. The shorter leg is opposite the 30 angle and the longer leg is
opposite the 60 angle"

Trigonometric Functions
The study of trigonometry probably began when early astronomers used the length of a shadow cast by a stick, called a gnomon, to determine the tide of day. As early as 1500 B.C. the Egyptians, along with astronomers in China, Mesopotamia, and India all understood that as the sun rises in the sky, it casts a unique shadow. In mathematical terms, the shadow is a function of the time of day.
The word trigonometry was derived from two Greek words – trigon meaning triangle and metra meaning measurement. So, trigonometry began as the study of the relationships between the angles and sides of a right triangle.

What is a Pythagorean triple?
A Pythagorean triple is a set of three positive whole numbers a, b, and c that are the lengths of the sides of a right triangle. This means that a, b, and c satisfy the equation from the Pythagorean Theorem, namely
a2 + b2 = c2.
The smallest example is a = 3, b = 4, and c = 5. You can check that
32 + 42 = 9 + 16 = 25 = 52.
Sometimes we use the notation (a,b,c) to denote such a triple.

Common Pythagorean Triples
3, 4, 5
6, 8, 10
5, 12, 13
7, 24, 25

m <A = 30º

15

c

Problem #1

Problem #2

Problem #4

Problem #3

Special
Triangles