Question

Prove analytically that in a right angeled triangle the mid - point of the hypotenuse is equidistant from the three angular points.

Answer 1

Choose the vertices of the right angled triangle as $O(0,0),A(a,0),B(0,b)$. Since the triangle is right angled

$\therefore A{B}^2=a^2+b^2$ or $\therefore AB=\sqrt{a^2+b^2}$

If C be the mid - point of AB , then

$CA=CB=\frac{1}{2}AB=\frac{1}{2}\sqrt{a^2+b^2}$

Also $C{O}^2=(a/2-0{)}^2+(b/2-0{)}^2$

$=\frac{1}{4}(a^2+b^2)$ so that

$CO=\frac{1}{2}\sqrt{a^2+b^2}$ Hence Prove