Question

In the figure, $O$ is the center, $PQ=RS$, and A and B are the midpoints of $PQ$ and $RS$, respectively. Which one of the following is true?

• A. $\angle OBS={90}^o$
• B. $\angle OAP={90}^o$
• C. $\angle OAB={90}^o$
• D. $\angle OAB=\angle OBA$

Answer 1

The correct option is D $\angle OAB=\angle OBA$

Given A, B are midpoints

$⟹PA=AQ=RB=BS=P$(since PQ = RS)

WKT
$OP=OQ=OR=OS=radius=r$

Now consider $△AOP$

$O{P}^2=A{P}^2+A{O}^2$

$⟹AO=\sqrt{r^2-p^2}$

In $△AOP$

$O{S}^2=O{B}^2+B{S}^2$

$⟹OB=\sqrt{r^2-p^2}$

$⟹AO=OB$

Since sides are equal, angles opposite sides will also be equal