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प्रश्न
In Fig.2, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively, If AB = 29 cm, AD = 23 cm, ∠B = 90° and DS = 5 cm, then the radius of the circle (in cm.) is:
पर्याय
(A) 11
(B) 18
(C) 6
(D) 15
उत्तर
We know that, the lengths of the tangents drawn from an external point to a circle are equal.
DS = DR = 5 cm
∴ AR = AD − DR = 23 cm − 5 cm = 18 cm
AQ = AR = 18 cm
∴ QB = AB − AQ = 29 cm − 18 cm = 11 cm
QB = BP = 11 cm
In right Δ PQB, PQ2 = QB2 + BP2 = (11 cm)2 + (11 cm)2 = 2 × (11 cm)2
PQ = `11sqrt2r` cm … (1)
In right ΔOPQ,
PQ2 = OQ2 + OP2 = r2 + r2 = 2 r2
PQ = `11sqrt2r`… (2)
From (1) and (2), we get
r = 11 cm
Thus, the radius of the circle is 11 cm.
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