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 ΔABC = 21 cm² then find the lengths of sides AB and AC.

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 = 4cmand CD = CF = 3cm
Now,
Area (ΔABC) = Area(ΔBOC)+ Area(ΔAOB) + Area(ΔAOC)

`⇒21 = 1/2 xx BC  xx OD +1/2 xx AB  xx OE +1/2 xx AC xx OF`

`⇒ 42 = 7 xx 2 + (4+x) xx 2 +(3+ x) xx2`

⇒ 21 = 7 + 4 + x + 3 + x

⇒ 21=14+2x

⇒ 2x  =7

⇒  x= 3.5 cm 

∴ AB = 4 + 3.5 = 7.5 cm and AC = 3 +3.5 = 6.5 cm