Question

A chord of a circle is $12$ cm which is at a distance of $8$ cm from center. Find the length of the chord of the same circle which is at a distance of $6$ cm from the centre.

• A. $20$ cm
• B. $24$ cm
• C. $16$ cm
• D. cm

Answer 1

Answer: (C)

$16$ cm

Let $O$ be the centre of the circle.

$AB=12cm$ ............ (Given)

The perpendicular distance of $AB$ from $O$ is $8cm$

$CD$ is another chord which is at a perpendicular distance of $6cm$ from $O$.

Join $OB$ and $OC$ which becomes radii of given circle.

Draw perpendiculars $OM$ and $ON$ on $AB$ and $CD$ respectively.

$\therefore OM=8cm$ and $ON=6cm$

Also, $\angle BMO={90}^o=\angle CNO$

$M$ is the midpoint of $AB$ ............ [Perpendicular from centre to any of its chord bisects the latter]

$\therefore BM=\frac{1}{2}AB=\frac{1}{2}\times 12cm=6cm$

Since, $\angle BMO={90}^o$

By Pythagoras theorem, we have

$OB=\sqrt{{OM}^2+{BM}^2}=\sqrt{8^2+6^2}cm=10cm=OC$

Since, $\angle CNO={90}^o$

$⟹CN=\sqrt{{OC}^2-{ON}^2}=\sqrt{{10}^2-6^2}cm=8cm$

$N$ is the midpoint of $CD$

$⟹CD=2CN=16cm$

Hence, option C is correct.