# Angle in Semicircle is a Right Angle

Consider the equation of the circle with the center at the origin $O\left( {0,0} \right)$ is given by

${x^2} + {y^2} = {r^2}\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Let $AOB$ be any diameter of the circle and $P\left( {{x_2},{y_2}} \right)$ be any point on the given circle.

We shall show that $m\angle APB = {90^ \circ }$.

Suppose that coordinates of $A$ are $\left( {{x_1},{y_1}} \right)$, then $B$ has coordinates $\left( { – {x_1}, – {y_1}} \right)$.

Slope of $AP = \frac{{{y_1} – {y_2}}}{{{x_1} – {x_2}}} = {m_1}$
Slope of $BP = \frac{{{y_1} + {y_2}}}{{{x_1} + {x_2}}} = {m_2}$

Now multiplying the slopes ${m_1}$and ${m_2}$, we get
${m_1}{m_2} = \frac{{{y_1}^2 – {y_2}^2}}{{{x_1}^2 – {x_2}^2}}\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Since points $A\left( {{x_1},{y_1}} \right)$ and $P\left( {{x_2},{y_2}} \right)$ lie on the circle, we have
$\left. \begin{gathered} {x_1}^2 + {y_1}^2 = {r^2} \Rightarrow {x_1}^2 = {r^2} – {y_1}^2 \\ {x_2}^2 + {y_2}^2 = {r^2} \Rightarrow {x_2}^2 = {r^2} – {y_2}^2 \\ \end{gathered} \right\}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)$

Substituting the values of ${x_1}^2$ and ${x_2}^2$ from equation (ii) into equation (i), we get
${m_1}{m_2} = \frac{{{y_1}^2 – {y_2}^2}}{{\left( {{r^2} – {y_1}^2} \right) – \left( {{r^2} – {y_2}^2} \right)}} = \frac{{{y_1}^2 – {y_2}^2}}{{ – \left( {{y_1}^2 – {y_2}^2} \right)}} = – 1$

This is the condition of perpendicular lines. Thus $AP \bot BP$ and so $m\angle APB = {90^ \circ }$.