Q.182. Geometry, in-radius.

Question 182.

In a triangle ABC, in-radius r = 4 cm. Let incircle touch the sides BC, CA and AB at D, E and F respectively. It is given that BD = 8 cm and DC = 6 cm. Find the sides AB and AC.

Answer 182.

Refer to the figure.

Let AF = AE = x cm

Also note that BF = BD = 8 cm and CE = CD = 6 cm.

Area of ΔABC

= area of (ΔIBC + ΔICA + IAB)

= (1/2) * 4 * (14 + 6 + x + 8 + x)
= 4(x + 14) ... (1)
 Semi-perimeter, s = x + 14 cm
=> area of ΔABC using Heron's formula
= √[s(s-a)(s-b)(s-c)]
= √[(x+14) * x * (8) * (6)]
= 4 √[3x(x+14)] ... (2)

From (1) and (2),
4 √[3x(x+14)] = 4(x+14)
=> √(3x) = √(x+14)
=> 3x = x+14
=> x = 7 cm
 => AB = x + 8 = 7 + 8 = 15 cm
and AC = x + 6 = 7 + 6 = 13 cm.

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