Question

Prove the following
Angle in a semi circle is a right angle.

Answer 1

Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be vectors that form and inscribed angle in a semi-circle.
Initial point $=(-a,0)$.
Terminal point $=(a\cos X,a\sin X)$
So, $\overrightarrow{u}=(|a|\cos X+a,|a|\sin X)$
and
$\overrightarrow{v}=(|a|\cos X-a,|a|\sin X)$
Taking, dot product
$\overrightarrow{u}.\overrightarrow{v}=$ $|a{|}^2{\cos }^{2}X-a^2+|a{|}^2{\sin }^{2}X$
$=|a{|}^2({\cos }^{2}X+{\sin }^{2}X)-|a{|}^2$
$=|a{|}^2-|a{|}^2=0$
Hence, the problem.