Question

Prove by vector method, that the triangle inscribed in a semi-circle is a right angle.

Answer 1

Let AB be diameter and O be the centre of a circle.

Let P be a point on the semi-circle.

Join PA,PB & PO.

By the law of triangle of vectors

vector PA= vector PO+ vector OA

vector PB= vector PO+vector OB= vector PO - vector OA

Since vector OB = vector OA

Consider,

$\begin{array}{c}vector\left(PA\cdot PB\right)=vector\left(PO+OA\right)\cdot vector\left(PO-OA\right)\\ =vector{\left|PO\right|}^2-vector{\left|OA\right|}^2\\ Since,vector\left|PO\right|=vector\left|OA\right|=radiusthecircle.\\ =0\\ ThereforevectorPAperpendiculartovectorPB\\ Therefore,\angle APB={90}^{\circ }\end{array}$