In the given figure, a triangle `ABC` is drawn to circuscribe a circle of radius `3cm`, such that the segments `BD` and `DC` into which `BC` is divided by the point of contact `D` are of lengths `6m` and `8cm` respectively. Find the side `AB`, if the area of `DeltaABC` is `63cm^(2)`.

Explanation

We know that the lengths of tangents drawn from an exterior point to be a circle are equal.
`:. AE=AF=x cm` (say)
`BD=BF=6cm`,
`CD=CE=8cm`.
And so, `AB=AF+BF=(x+6)cm`, `BC=BD+CD=14cm`,
`CA=CE+AE=(x+8)cm`
Join `OE` and `OF` and also `OA`, `OB` and `OC`.
`:.ar(DeltaABC)=ar(DeltaOAB)+ar(DeltaOBC)+ar(DeltaOCA)`
`implies63=((1)/(2)xxABxxOF)+((1)/(2)xxBCxxOD)+((1)/(2)xxCAxxOE)`
`implies63={(1)/(2)xx(x+6)xx3}+((1)/(2)xx14xx3)+{(1)/(2)xx(x+8)xx3}`
`implies63=(3)/(2)xx(2x+28)impliesx=7`.
`:. AB=(x+6)cm=(7+6)cm=13cm`.