Question

Two chords AB, CD of lengths 16 cm and 30 cm, are parallel. If the distance between AB and CD is 23 cm. Find the radius of the circle.

Answer 1

Let AB and CD be the two parallel chords of a circle with centre O and radius r cm.
OM ⊥ AB and ON ⊥ CD
AM = ${\displaystyle \frac{1}{2}\text{AB}=\frac{1}{2}\times 16}$ = 8 cm

CN = ${\displaystyle \frac{1}{2}\text{CD}=\frac{1}{2}\times 30}$ = 15 cm

Let OM = x cm, MN = 23 cm
so ON = (23 - x) cm
OA = OC = r cm

In Δ OAM,
OA² = AM² + OM²
⇒ r² = (8)² + (x)²            ....(i)

In Δ OCN,
OC² = CN² + ON²
⇒ r² = (15)² + (23 - x )²      ....(ii)
From (i) and (ii),
x² + 64 = 225 + (23 - x)²
⇒ x² + 64 = 225 + 529 - 46x + x²
⇒ 46x = 225 + 529 - 64
⇒ 46x = 690
⇒  x = 15 cm
From (i),
r² = (8)² + (15)²
r² = 64 + 225
r² = 289
⇒ r = 17 cm
Hence, the radius of a circle is 17 cm.