Hypotenuse of a Right Triangle

Right Triangle: A right triangle is a triangle in which one of the angle is 90°.
 


In a right triangle, one side is the base, the longest side is called the hypotenuse and the third side is the height of the triangle.

Pythagorean Theorem: The sum of the squares of the two sides (base and height) is equal to the square of the hypotenuse.
 
In the above right triangle, the base is denoted by ‘b’, height by ‘a’ and the hypotenuse by ‘c’.

Applying the Pythagorean theorem, we get
          a² + b² = c²
  [(height)² + (base)² = (hypotenuse)² ]  (hypotenuse of a right triangle formula)

Pythagorean Theorem Statement: In a right triangle, the square of the hypotenuse is equal to the squares of the sum of the other two sides

Proof: We are given a right triangle with right angle at B
We need to prove that: AC² = AB² + BC²
 Let us draw a perpendicular, BD $\perp$  AC

Now,  $\bigtriangleup$ ADB is similar to $\bigtriangleup$ ABC

(According to the theorem, If a perpendicular is drawn from the vertex at the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other).

So,   $\frac{AD}{AB}$ = $\frac{AB}{AC}$ (sides are proportional)
      
                     AD. AC = AB²             -------------------(1)
 
Also, $\bigtriangleup$ BDC is similar to $\bigtriangleup$ ABC
       
      $\frac{CD}{BC}$ = $\frac{BC}{AC}$
                  
CD. AC = BC²                 -------------------(2)

Adding (1) and (2)
AD. AC + CD.AC = AB² + BC²
AC (AD + CD) = AB² + BC²
AC.AC = AB² + BC²
AC² = AB² + BC²
Hence the Pythagorean Theorem is proved.