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.