ABCD is a quadrilateral such that ∠D = 90°. A circle (O, r) touches the sides AB, BC, CD and DA at P,Q,R and If BC = 38 cm, CD = 25 cm and BP = 27 cm, find r.
Since tangent to a circle is perpendicular to the radius through the point.
∴ ∠ORD = ∠OSD = 90°
It is given that ∠D = 90° Also, OR = OS. Therefore, ORDS is a square.
Since tangents from an exterior point to a circle are equal in length.
∴ BP = BQ
CQ = CR and DR = DS
Now, BP = BQ
⇒ BQ = 27 [∵BP = 27 cm (Given)]
⇒ BC – CQ = 27
⇒ 38 – CQ = 27 [∵BC = 38 cm ]
⇒ CQ = 11cm
⇒ CR = 11cm [∵CR = CQ ]
⇒ CD – DR = 11
⇒ 25 – DR = 11 [∵CD = 25cm ]
⇒ DR = 14 cm
But, ORDS is a square.
Therefore, OR = DR = 14 cm
Hence, r = 14 cm
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