Question

Using vectors, prove that angle in a semicircle is a right angle.

Answer 1

Let $O$ be the centre of the semi-circle and $BA$ be the diameter. Let $P$ be any point on the circumference of the semi-circle.
Let $\overrightarrow{OA}=\overrightarrow{a}$ then $\overrightarrow{OB}=-\overrightarrow{a}$

Let $\overrightarrow{OP}=\overrightarrow{r}$
$\therefore \overrightarrow{AP}=\overrightarrow{OP}-\overrightarrow{OA}=\overrightarrow{r}-\overrightarrow{a}$

$\overrightarrow{BP}=\overrightarrow{OP}-\overrightarrow{OB}=\overrightarrow{r}-(-\overrightarrow{a})=\overrightarrow{r}+\overrightarrow{a}$

$\overrightarrow{AP}.\overrightarrow{BP}=(\overrightarrow{r}-\overrightarrow{a}).(\overrightarrow{r}+\overrightarrow{a})$

$=\overrightarrow{r^2}-\overrightarrow{a^2}$

$=\overrightarrow{a^2}-\overrightarrow{a^2}$ $[\because r=a$ as, $OP=OA]$
$=0$

$\because \overrightarrow{AP}$ is perpendicular to $\overrightarrow{BP}$
Hence Proved. $\Rightarrow \angle APB={90}^o$