Question

prove that the angle inscribed in a semicircle is a right angled triangle​

Answer 1

Answer:

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,

vector(PA⋅PB)=vector(PO+OA)⋅vector(PO−OA)=vector∣PO∣2−vector∣OA∣2Since,vector∣PO∣=vector∣OA∣=radiusthecircle.=0ThereforevectorPAperpendiculartovectorPBTherefore,∠APB=90∘