Question : In the given figure AB = 30 cm and CD = 24 cm. What is the value (in cm) of MN?

Option 1: 18

Option 2: 9

Option 3: 12

Option 4: 15

Correct Answer: 18

Solution :

We have, AB = 30 cm and CD = 24 cm.
Construction: Let O be the centre for both circles. And draw a line OA, OD, OM and LP.
Such that OP and OL are perpendicular on AB and MN.
From the figure,
⇒ CP = PD = 12 cm
⇒ AP = PB = 15 cm
In ΔOPD by Pythagoras theorem,
⇒ OD ² = OP ² + PD ²
⇒ OD ² = OP ² + 12 ²
⇒ OD ² – OP ² = 12 ² ____(i)
In ΔOPA by Pythagoras theorem,
⇒ OA ² = OP ² + AP ²
⇒ OA ² = OP ² + 15 ²
⇒ OA ² – OP ² = 15 ² ____(ii)
From equations (i) and (ii),
OA ² – OD ² = 225 – 144 = 81 ____(iii)
In ΔOLM by Pythagoras theorem,
OD = OL and OA = OM ____(iv)
⇒ OM ² = OL ² + LM ²
From (iv)
⇒ OA ² = OD ² + LM ²
⇒ OA ² – OD ² = LM ² ____(v)
On comparing equations (iii) and (v),
⇒ LM ² = 81 cm
⇒ LM = 9 cm
⇒ MN = 2LM
⇒ MN = 2 × 9 = 18 cm
Hence, the correct answer is 18.