Question

In the figure, $AB$ and $PQ$ are chords of a circle with center $O.$ Point $M$ lies on $AB$ such that $OM$ is perpendicular to $AB,$ and point $N$ lies on $PQ$ such that $ON$ is perpendicular to $PQ.$ Find the length of $PQ.$

• A. $4$ cm
• B. $8$ cm
• C. $10$ cm
• D. $5$ cm

Answer 1

The correct option is B $8$ cm

Given, $OM=4$ cm, $AB=6$ cm, $ON=3$ cm
$ON\perp PQ,OM\perp AB$

We know that,
the perpendicular from the center of a circle to a chord bisects the chord
$\therefore BM=3(OM\perp AB)$
$\angle OMB={90}^{\circ }$

To find the length of PQ, we need to determine the radius of a circle.

Find radius using pythagorean theorem,
$O{M}^2+M{B}^2=O{B}^2\Rightarrow 4^2+3^2=O{B}^2\Rightarrow O{B}^2=25\Rightarrow OB=5$

Find the length of NP using Pythagorean Theorem in $△ONP,$ we get
$O{N}^2+N{P}^2=O{P}^2$
$\Rightarrow 3^2+N{P}^2=5^2(\because OP=OB=5)$
$\Rightarrow N{P}^2=16\Rightarrow NP=4$
$\therefore PQ=NP+NQ$
$PQ=4+4$ $(\because NP=NQ)$
$PQ=8$