Question

Two concentric circles are of radii $5cm$ and $3cm$. Find the length of the chord of the larger circle which touches the smaller circle.

Answer 1

Connecting OP, OA and OB

$OP=3cm$

$OA=OB=5cm$

Since AB is tangent to $C_1$

$OP\perp AB$ [Tangent at any point of circle is perpendicular to the radius through point of contact]

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

Using Pythagoras theorem

In right triangle OAP

$O{A}^2=O{P}^2+A{P}^2$

$5^2=3^2+A{P}^2$

$25=9+A{P}^2$

$A{P}^2=16$

$AP=4cm$

In right triangle OPB

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

$5^2=3^2+A{P}^2$

$25=9+P{B}^2$

$P{B}^2=16$

$PB=4cm$

Hence

$AB=AP+PB$

$=4+4$

$=8cm$