(i) A circle is inscribed in a ABC having sides BC, CA and AB 16 cm, 20 cm and 24 cm respectively as shown in the figure Find AD, BE and CF.

(ii) If AF=4cm, BE=3cm, AC=11cm, then find BC.

i) Tangents drawn from external point are equal

AD and AF are tangents from point A

⇒ AD = AF = a

BF and BE are tangents from point B

⇒ BD = BE = b

CD and CE are tangents from point C

⇒ CF = CE = c

From figure

We have AC = AF + FC

⇒ 20 = a + c …(i)

Also, AB = AD + DB

⇒ 24 = a + b …(ii)

And CB = CE + EB

⇒ 16 = c + b …(iii)

Add (i), (ii) and (iii)

⇒ 20 + 24 + 16 = a + c + a + b + c + b

⇒ 60 = 2(a + b + c)

⇒ a + b + c = 30 …(iv)

Substitute (i) in (iv)

⇒ 20 + b = 30

⇒ b = 10

Substitute (ii) in (iv)

⇒ 24 + c = 30

⇒ c = 6

Substitute (iii) in (iv)

⇒ 16 + a = 30

⇒ a = 14

Hence AD = a = 14 cm, BE = b = 10 cm and CF = c = 6 cm

ii)

Tangents drawn from external point are equal

CF and CE are tangents from point C

⇒ CF = CE = c

From figure

AC = AF + FC

⇒ 11 = 4 + c

⇒ c = 7 cm

Hence EC = c = 7 cm

We have BC = BE + EC

⇒ BC = 3 + 7

⇒ BC = 10 cm

Hence BC is 10 cm