In the adjoining figure, quadrilateral ABCD is circumscribed. If the radius of the in circle with centre O is 10 cm and AD DC, find the value of x.

In the given figure,


DS and DR are the two tangents drawn from an external point D at the point of contacts S and R respectively. And,


OS DS and OR DR


[ radius of a circle is always to the tangent at the point of contact.]


OSDR is a square [ AD DC (Given)]


DR = 10 cm


Similarly,


BA and BQ are the two tangents drawn from an external point B at the point of contacts A and Q respectively.


BA = BQ = 27 cm


[ Tangents drawn from an exterior point to the circle are equal in length]


QC = BC – BQ = 38 – 27 = 11 cm


Also, CR and CQ are the two tangents drawn from an external point C at the point of contacts R and Q respectively.


CR = CQ = 11 cm


[ Tangents drawn from an exterior point to the circle are equal in length]


DC = x = DR + CR = 10 + 11 = 21 cm.


Thus, the value of x is 21 cm.


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