Question

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BC and DC into which BC is divided by the point of contact D, are of
lengths 6cm and 9cm respectively. If the area of 2 ΔABC = 54cm² 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 me same external point are congruent
Now, we have
AE = AF, BD = BE =6 cm and CD = CF = 9 cm
Now,

Area(ΔABc) =  Area ( ΔBOC)+ Area( ΔAOB) +Area( ΔAOC)

`⇒ 54 = 1/2 xx BC xx OD + 1/2 AB xx OE + 1/2 xx  AC xx  OF `

⇒ 108 = 15 × 3+(6+x) ×3 +(9+x) × 3

⇒ 36=15+6+x+9+x

⇒ 36=30+2x

⇒ 2x = 6

⇒ x = 3 cm 

∴ AB =6+3=9 cm and AC =9+3=12cm