Question:
ABC is a right-angled trianle and O is the mid-point of the side opposite to the right angle. Explain why O is equidistant from A, B and C.
Solution:
(i) Construct a triangle $\mathrm{ABC}$ right angle at $\mathrm{B}$.
(ii) Suppose $\mathrm{O}$ is the mid point of $\mathrm{AC}$.
(iii) Complete the rectangle $\mathrm{ABCD}$ having $\mathrm{AC}$ as its diagonal.
Since diagonals of a rectangle are equal and they bisect each other, $\mathrm{O}$ is the midpoint of both $\mathrm{AC}$ and $\mathrm{BD}$.
$\therefore \mathrm{OA}=\mathrm{OB}=\mathrm{OC}$