Question

If radii of two concentric circles are $4$ cm and $5$ cm, then the length of each chord of one circle which is tangent to the circle is

• A. $3$ cm
• B. $6$ cm
• C. $9$cm
• D. $1$ cm

Answer 1

Answer: (B)

$6$ cm

Here. above figure shows two concentric circles of radius $4cm$ and $5cm$

Let $AB$ be the chord of the larger circle which is tangent to the smaller circle at point $P.$

Joining $OA,OB$ and $OP$

$OA=OB=5cm$

$OP=4cm$

We know that tangent is perpendicular to the radius.

$\therefore$ $OP\perp AB$

$\therefore$ $\angle OPA=\angle OPB={90}^o$

In right-angled $△OAP$

$\Rightarrow$ $O{A}^2=O{P}^2+A{P}^2$ [ By Pythagoras theorem ]

$\Rightarrow$ $(5{)}^2=(4{)}^2+A{P}^2$

$\Rightarrow$ $25=16+A{P}^2$

$\Rightarrow$ $A{P}^2=9$

$\therefore$ $AP=3cm$

In right-angled $△OPB$

$\Rightarrow$ $O{B}^2=O{P}^2+B{P}^2$

$\Rightarrow$ $5^2=4^2+B{P}^2$

$\Rightarrow$ $25=16+B{P}^2$

$\Rightarrow$ $B{P}^2=9$

$\therefore$ $BP=3cm$

$\Rightarrow$ $AB=AP+PB$

$=3+3$

$=6cm$