Question

If two parallel chords on the same side of the centre of a circle are $6$cm and $8$cm, and they are $1$cm apart, then the diameter of the circle will be

• A. $14$cm
• B. $10$cm
• C. $8$cm
• D. $5$cm

Answer 1

The correct option is B $10$cm

Let the length of two parallel chords $AB$ and $CD$ on the same side of the centre O of a circle be $6$cm and $8$cm respectively. Thy are $1$cm apart from each other.
The perpendicular drawn from the centre will bisect them at the points M and N respectively.
$\therefore AN=BN$
$=\frac{1}{2}AB=3$cm
and $CM=DM=\frac{1}{2}CD$
$=4cm$
Let the radius of the circle be R cm $=OD$
$=OB$
and $OM=xcm$
$\because O{D}^2=O{M}^2+M{D}^2$
$\Rightarrow R^2=x^2+4^2$ ....(1)
and $O{B}^2=ON62+N{B}^2$
$\Rightarrow R62=(x+1{)}^2+3^2$ ....(2)
$(x+1{)}^2+3^2=x^2+4^2$
$\Rightarrow (x+1)-x^2-x^2=4^2-3^2$
$\Rightarrow 2x+1=(4+3)(4-3)$
$=7$
$\therefore x=\frac{7-1}{2}$
$=3$
and $R^2=3^2+4^2$
$=5^2$
$R=5cm$

$\therefore$ Diameter of the circle $=2R=2\times 5$$=10cm$