Pythagorean theorem states that for a right triangle, the square of its hypotenuse is equal to the sum of squares of its adjacent arms. Mathematically it can be expressed from the following figure as

c^2 = a^2 + b^2 \\

Proof: From the trigonometry we know,

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

or, \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin 90}

or, \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{1}

or, a = c\sin A

and \quad b = c\sin B

So, \quad a^2 + b^2 = c^2(\sin^2 A + \sin^2 B)

= c^2(\sin^2 A + \sin^2 (90-A)) \quad [since A + B = 90]

= c^2(\sin^2 A + \cos^2 A) \\

= c^2

Example: Suppose that you walked in the north 3 miles and then in the west 3 miles. What is the distance between your starting point and ending point?

From the figure,

                BC = 3, AB = 3

AC^2 = AB^2 + BC^2 \\

= 9 + 9 \\

= 18 \\

So, AC = \sqrt{18} = 4.24 miles

Example: From the following figure, determine the value of OA.

From right triangle OBC,

OC^2 = OB^2 + BC^2

or, BC^2 = OC^2 - OB^2

= 17^2 - 13^2

= 120

or, BC = 10.95

From right triangle OAB,

OA^2 = OB^2 + AB^2

or, OA^2 = OB^2 + (AC+BC)^2

= 13^2 + (7 + 10.95)^2

or, OA^2 = 491.2

or, OA = 22.16