Subject: Geometry

Pythagorean Theorem

During the reign of ancient Greece, a mathematician and philosopher named Phytagoras made an interesting discovery about right triangles. In this work, he discovered important relationship between the three sides of a right triangle. This discovery made Phytagoras very famous and the feat was even named before him - The Phytagorean Theorem.

Phytagoras discovered that when you made a square on the three sides of a right triangle, the area of the biggest square is equal to the sum of the other two squares' areas. This is illustrated in the figure below. The area of the blue square is the same when we add the area of the red and green squares.

Formal Definition of Phytagorean Theorem

According to the Wikipedia, Phytagorean theorem is stated in the following:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

This theorem would be useless without the representation of a mathematical equation. Since the area of square is obtained by squaring the length of one side. Then according to the figure above, the area of blue square is A^2, the red square is B^2 and C^2 for the green square. Thus, the mathematical expression of the Phytagorean Theorem is

A^2 = B^2 + C^2

which is also called the Phytagorean Equation.

This theorem is of fundamental importance in Geometry because not just it stresses the relationship between sides in right triangles but it also serves as a basis for the definition of distance between two points. However, one must remember that Phytagorean Theorem only applies to right triangles. For other forms of triangles, we have the law of sines and cosines which interestingly enough, were discovered with the aid of the Phytagoras' Theorem.

Example #1

Who discovered the Phytagorean Theorem?
a.) Newton
b.) Einstein
c.) Phytagoras
d.) Phyton

c.) Phytagoras

Example #2

If the set of numbers below constitute the sides of a triangle (the largest being the hypotenuse), which set represents a right triangle?
a.) 1,2,3
b.) 3,4,5
c.) 4,5,6

b.) 3,4,5

Proof: Using Phytagorean Theorem:

5^2=4^2+3^2
25= 16+9
25=25

Example #3

You are travelling 3 km north and turns right for another 2 km. How far are you from the start of your travel?
a.) 5km
b.) 6km
c.) 3.6 km
d.) 2.5 km

c.) 3.6 km

Proof: The distance you travelled is 2+3=5km but that's not how far you are from the starting point. To determine how far you are, you may use Phytagorean Theorem since your journey forms a right triangle with respect to your starting point. Using Phytagorean Theorem:

x^2=2^2+3^2
x^2= 4+9
x=\sqrt{13}
x=3.6

Example #4

A right triangle has sides 3cm, 7cm and 7.6 cm. What is its area?
a.) 12.8cm
b.) 10.5cm
c.) 53.2cm

b.) 10.5cm

Since the area of a triangle is half the base times height.

Example #5

What is the diagonal distance across a square of area 4?
a.) 2.8
b.) 2.1
c.) 3.2

a.) 2.8

The diagonal is just the hypotenuse of a right triangle whose are 2. Using the Phytagorean Equation, we get the answer.