Question

In figure 11, $OP$ bisects $\angle AOC$, OQ bisects $\angle BOC$ and $OP\perp OQ$. Show that the points $A,O$ and $B$ are collinear.

Answer 1

Given:

$OP$ bisects $\angle AOC$, $OQ$ bisects $\angle BOC$ and $OP\perp OQ$

To prove: The points $A,O$ and $B$ are collinear.

Proof:

Since, $OP$ bisects $\angle AOC$,

$\angle AOP=\angle COP$ …… (1)

Since, $OQ$ bisects $\angle BOC$,

$\angle BOQ=\angle COQ$ …… (2)

Now, $\angle AOB$

$=\angle AOP+\angle COP+\angle COQ+\angle BOQ$

$=\angle COP+\angle COP+\angle COQ+\angle COQ$

From (1) and (2), we have

$=2(\angle COP+\angle COQ)$

$=2\angle POQ$

$=2\left(90{}^{\circ }\right)$

$=180{}^{\circ }$

Therefore, points $A,O$ and $B$ are collinear.