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.

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Construct}\mathrm{a}\mathrm{triangle}\mathrm{ABC}\mathrm{right}\mathrm{angle}\mathrm{at}\mathrm{B}. \\ \left(\mathrm{ii}\right)\mathrm{Suppose}\mathrm{O}\mathrm{is}\mathrm{the}\mathrm{mid}\mathrm{point}\mathrm{of}\mathrm{AC}. \\ \left(\mathrm{iii}\right)\mathrm{Complete}\mathrm{the}\mathrm{rectangle}\mathrm{ABCD}\mathrm{having}\mathrm{AC}\mathrm{as}\mathrm{its}\mathrm{diagonal}. \\ \mathrm{Since}\mathrm{diagonals}\mathrm{of}\mathrm{a}\mathrm{rectangle}\mathrm{are}\mathrm{equal}\mathrm{and}\mathrm{they}\mathrm{bisect}\mathrm{each}\mathrm{other},\mathrm{O}\mathrm{is}\mathrm{the}\mathrm{midpoint}\mathrm{of}\mathrm{both}\mathrm{AC}\mathrm{and}\mathrm{BD}. \\ \therefore \mathrm{OA}=\mathrm{OB}=\mathrm{OC} \end{gathered}$$