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^2Example: 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^2or, BC^2 = OC^2 - OB^2
= 17^2 - 13^2= 120
or, BC = 10.95
From right triangle OAB,
OA^2 = OB^2 + AB^2or, OA^2 = OB^2 + (AC+BC)^2
= 13^2 + (7 + 10.95)^2or, OA^2 = 491.2
or, OA = 22.16