Question

A ∆ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if it is given that ar(∆ABC) = 90 cm².

Answer 1

Given, ∆ABC circumscribes a circle of radius 4 cm.

Segments on side BC:

BD = 10 cm

CD = 8 cm

∴ BC = BD + DC

= 10 + 8

= 18 cm

Let tangent lengths from vertex A be:

AF = AE = x

Because tangents from an external point to a circle are equal.

Tangents from B:

BD = BE = 10

Tangents from C:

CD = CF = 8

Therefore, the sides of the triangle:

AB = AF + BE

= x + 10

AC = AE + CF

= x + 8

Since O is the incenter, it divides △ABC into three triangles whose sum of areas equals the area of the whole triangle.

ar(△ABC) = ar(△AOC) + ar(△BOC) + ar(△AOB)

`90 = 1/2 xx r xx AC + 1/2 xx r xx BC + 1/2 xx r xx AB`

`90 = 1/2 xx 4(x + 8 + 18 + x + 10)`

90 = 2(2x + 36)

90 = 4x + 72

4x = 18

x = `18/4`

x = 4.5 cm

∴ AB = x + 10

= 4.5 + 10

= 14.5 cm

AC = x + 8

= 4.5 + 8

= 12.5 cm