Question

In the figure, $O$ is the centre of the circle of radius $5\text{cm}.$ $P$ and $Q$ are points on chords $AB$ and $CD$ respectively such that $OP\perp AB,OQ\perp CD,AB||CD,AB=6\text{cm}$ and $CD=8cm.$ Determine $PQ$.

• A. $5cm$
• B. $6cm$
• C. $7cm$
• D. $8cm$

Answer 1

The correct option is C

$7cm$

Given, $AB=6\text{cm}$.
Then, $AP=\frac{AB}{2}=3\text{cm}$ and $CQ=\frac{CD}{2}=4\text{cm}$
($\because$ Perpendicular from the centre of a circle to a chord bisects the chord)
$OA=OC=\text{Radius}=5\text{cm}$
$\Rightarrow OP=\sqrt{\left({OA}^2\right)-\left({AP}^2\right)}$ [Pythagoras' theorem]
$\Rightarrow OP=\sqrt{5^2-3^2}=\sqrt{16}=4\text{cm}$
Similarly, $OQ=\sqrt{O{C}^2-C{Q}^2}$
$\Rightarrow OQ=\sqrt{5^2-4^2}=\sqrt{9}=3\text{cm}$
Therefore, $PQ=PO+OQ=4+3=7\text{cm}$