Question

For the given concentric circles with radius 5 cm and 13 cm respectively. Find the length of the chord AB.

Answer 1

Join $OP$ and $OB$.
Given: $OA=OB=13cm$ [radiI of the outer circle]
and $OP=5cm$ [radius of the inner circle]
By Theorem- Tangent at any point is perpendicular to the radius through the point of contact.
$\because$ Chord AB is a tangent to inner circle and OP is a radius of inner circle
$\therefore OP\perp AB$
Now, consider right $△$ $OPB$,
Applying pythagoras theorem,
$O{P}^2+P{B}^2=O{B}^2$
$\Rightarrow 5^2+P{B}^2={13}^2$
$\Rightarrow PB=\sqrt{169-25}$
$\Rightarrow PB=12cm$
By Theorem - In two concentric circle, the chord of outer circle that touches the inner circle is bisects at the point of contact with inner circle.
$\therefore AP=PB=12cm$
$\Rightarrow AB=AP+PB=12+12=24cm$
$\therefore$ Length of chord $AB=24cm$