In Fig. 3, the radius of in circle of ∆ABC of area 84 cm² is 4 cm and the lengths of the segments AP and BP into which side AB is divided by the point of contact P are 6 cm and 8 cm. Find the lengths of the sides AC and BC.

(Fig. 3)

Question

Philoid · Accepted Answer

Given: Area of ∆ABC = 84 cm²

PB = 8 cm

PA = 6 cm

AB = 14 cm

BQ = 8 cm (Common tangent with PB)

AR = 6 cm (Common tangent with PA)

Since QC & CR are Common tangent

Let QC = CR = x

AC = 6 + x

BC = 8 + x

So ΔROA and ΔPOA are congruent

ΔPOB and ΔQOB are congruent

ΔQOC and ΔROC are congruent

Area of ΔROA = ΔPOA = 12 cm²

Area of ΔPOB = ΔQOB = 16 cm²

Area of ΔQOC = ΔROC = 2x

The sum areas of all the small triangles = Area of ∆ABC

⇒ 2 × 12 + 2 × 16 + 2 × 2x = 84

⇒ 4x = 28

⇒ x = 7 cm

Answer: Length of AC = 13 cm & Length of BC = 15 cm

In Fig. 3, the radius of in circle of ∆ABC of area 84 cm2 is 4 cm and the lengths of the segments AP and BP into which side AB is divided by the point of contact P are 6 cm and 8 cm. Find the lengths of the sides AC and BC.(Fig. 3)