Question

an angle inscribed in a semicircle is a right angle triangle,prove by vector method.​

Answer 1

❤️Qúéßtîóñ:❤️⤵️

An angle inscribed in a semicircle is a right angle triangle,prove by vector method.

Àñßwër ❤️:⤵️

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 ∣ OAb∣ 2

Since,vector∣PO∣=vector∣OA∣=radiusthecircle.

=0

ThereforevectorPAperpendiculartovectorPB

Therefore,∠APB=90∘