Question

In figure, $O$ is the centre of the circle of radius $5cm,OP\perp AB,OQ\perp CD,AB||CD,\overline{AB}=6cm,\overline{CD}=8cm$. Determine $PQ$

• A. $5cm$
• B. $7cm$
• C. $6cm$
• D. $4cm$

Answer 1

The correct option is B $7cm$

Since the perpendicular from the centrer of the circle to a chord bisects the chord.
$\therefore P$ and $Q$ are the mid points of $AB$ and $CD$
$AP=\frac{1}{2}=AB==3cm$
$CQ=\frac{1}{2}CD=4cm$
In right triangles $OAP$ and $OCQ$, we have
$O{A}^2=O{P}^2+A{P}^2$ and $O{C}^2=O{Q}^2+C{Q}^2$
$5^2=O{P}^2+3^2$ and $5^2=O{Q}^2+4^2$
$\Rightarrow O{P}^2=5^2-3^2$ and $\Rightarrow O{Q}^2=5^2-4^2$
$O{P}^2=16$ and $O{Q}^2=9$
$\therefore OP=4$ and $\therefore OQ=3$
$\therefore PQ=OP+OQ=4+3=7cm$