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प्रश्न
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.
उत्तर
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 = `1/2"AB" = 1/2 xx 16` = 8 cm
CN = `1/2"CD" = 1/2 xx 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.
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