Question

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D are, of lengths 4 cm and 3 cm respectively. If the area of △ABC = 21 cm² then find the lengths of sides of AB and AC. [CBSE 2011]

Answer 1

Construction: Join OA, OB, OC, OE ⊥ AB at E and OF ⊥ AC at F

We know that tangent segments to a circle from the same external point are congruent.
Now, we have
AE = AF, BD = BE = 4 cm and CD = CF = 3 cm
Now,
$$\begin{gathered} \mathrm{Area}\left(△\mathrm{ABC}\right)=\mathrm{Area}\left(△\mathrm{BOC}\right)+\mathrm{Area}\left(△\mathrm{AOB}\right)+\mathrm{Area}\left(△\mathrm{AOC}\right) \\ \Rightarrow 21=\frac{1}{2}\times \mathrm{BC}\times \mathrm{OD}+\frac{1}{2}\times \mathrm{AB}\times \mathrm{OE}+\frac{1}{2}\times \mathrm{AC}\times \mathrm{OF} \\ \Rightarrow 42=7\times 2+\left(4+x\right)\times 2+\left(3+x\right)\times 2 \\ \Rightarrow 21=7+4+x+3+x \end{gathered}$$

$$\begin{gathered} \Rightarrow 21=14+2x \\ \Rightarrow 2x=7 \\ \Rightarrow x=3.5\mathrm{cm} \end{gathered}$$
∴ AB = 4 + 3.5 = 7.5 cm and AC = 3 + 3.5 = 6.5 cm