A circle is inscribed in a triangle ABC. It touches side AB at point X

SOLUTION

Let CY = CZ = x cm (Tangents drawn from a common point)

In ΔABC,

BX = BY & AX = AZ (Tangents drawn from a common point)

Apply the cosine rule,

Cos60˚ =

96 + 16x = 256 + 36 + 12x + x² – 100 – 20x – x²

96 + 16x = 192 – 8x

24x = 96

x = 4 cm

Now, side AC = 6 + x = 6 + 4 = 10 cm,

BC = 10 + x = 10 + 4 = 14 cm

Semi-perimeter of ΔABC =  = 20 cm

Area of ΔABC =

=

=

= 40 cm²

Radius of in-circle =  =  = 2

Area of In-circle =  = 12

Hence, required area = Area of ΔABC – Area of In-circle

= 40 – 12 cm²