In Fig. 4, a circle is inscribed in a ΔABC having sides BC = 8 cm, AB = 10 cm and AC = 12 cm. Find the lengths BL, CM and AN.

Let AN = x cm, CM = z cm and BL = y cm

As we know that,

Tangents from an external point to a circle are equal,

In given Figure we have

AN= AM = x [Tangents from point A]

BN = BL = y [Tangents from point B]

CL = CM = z [Tangents from point C]

Now, Given: AC = 12 cm

AM + MC = 12

x + z = 12

z = 12 – x…. [1]

and BC = 8 cm

BL + LC = 8

y + z = 8

z = 8 – y …… [2]

and

AB = 10 cm

AN + BN = 10

x + y = 10 …. [3]

equate [1] and [2] to get,

12 – x = 8 – y

4 = x - y … [4]

Add [3] and [4] to get,

x + x + y – y = 10 + 4

2x = 14

x = 7 cm

put value of x in [4] to get,

4 = 7 – y

y = 3 cm

put value of y in [2] to get,

z = 8 – 3

= 5 cm

Hence,

AN = 7 cm

CM = 5 cm

BL = 3 cm