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In Fig a circle is inscribed in a quadrilateral ABCD in which \(\angle B=90^\circ.\) If AD = 23 cm, AB = 29 cm and DS = 5 cm, find the radius r of the circle.

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Given: ABCD is a quadrilateral in which

\(\angle B=90^\circ\)

\(AD=23cm, DS=5cm\,and\,AB=29cm\)

Let the radius of the incircle be r cm

AD = DS = 5cm (tangent from an external point) 

Since AD=23 cm 

So, 

AR + RD + AD 

AR + 5 = 23 cm 

AR = 18 cm ------(i) 

and AQ = AR 

since AR = 18 cm 

So, AQ + QB = AB

Now OP and OQ are radius of the circle. So from tangent P and Q

\(\angle OPB=\angle OQB=90^\circ\)

\(OPBQ\) is a square

\(OP=QB\)

Radius of the circle = 11cm

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