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# In Quadrilateral Abcd; Angles D = 90°, Bc = 38 Cm and Dc = 25 Cm. a Circle is Inscribed in this Quadrilateral Which Touches Ab at Point Q Such that Qb = 27 Cm, Find the Radius of the Circle. - ICSE Class 10 - Mathematics

ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

#### Question

In quadrilateral ABCD; angles D = 90°, BC = 38 cm and DC = 25 cm. A circle is inscribed in this quadrilateral which touches AB at point Q such that QB = 27 cm, Find the radius of the circle.

#### Solution

BQ and BR are the tangents from B to the circle.
Therefore, BR =BQ = 27 cm.
Also RC = (38 −; 27) = 11cm
Since CR and CS are the tangents from C to the circle
Therefore, CS = CR = 11 cm
So, DS = (25 − 11) = 14 cm
Now DS and DP are the tangents to the circle
Therefore, DS = DP
Now, ∠PDS = 90° (given)
and OP  ⊥ AD, OS ⊥  DC
therefore, radius = DS = 14 cm

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Solution In Quadrilateral Abcd; Angles D = 90°, Bc = 38 Cm and Dc = 25 Cm. a Circle is Inscribed in this Quadrilateral Which Touches Ab at Point Q Such that Qb = 27 Cm, Find the Radius of the Circle. Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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