Question

Two circles of radii 5 cm and 3 cm are concentric. Calculate the length of a chord of the outer circle which touches the inner circle.

• A. 12 cm
• B. 2 cm
• C. 8 cm
• D. 23 cm

Answer 1

The correct option is C

8 cm

Given - Two concentric circle with radius 5 cm and 3 cm with centre O. PQ is the chord of the outer circle which touches the inner circle at L.

Construction: Join OL and OP.

So, OL = 3 cm, OP = 5 cm.
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
$\therefore \angle OLP={90}^{\circ }[\because \text{PQ is a tangent to inner circle]}$

$\text{In right}\Delta OLP,\text{applying pythagoras}\text{theorem,}$

$O{P}^2=O{L}^2+L{P}^2\Rightarrow (5{)}^2=(3{)}^2+L{P}^2\Rightarrow 25=9+L{P}^2\Rightarrow L{P}^2=25-9=16\therefore LP=\sqrt{16}=4cm$

Since, the radius remains the same, the length of OQ = 5 cm, OL = 3 cm. Hence, by Pythagoras theorem, LQ will also be 4 cm.

$\text{Hence,}PQ=2LP=2\times 4=8cm$

So, length of the chord is 8 cm.