# In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at point D,

**Question:**

In the given figure, a circle inscribed in a triangle ABC, touches the sides AB, BC and AC at point D, E and F respectively. If AB = 14 cm, BC = 8 cm and AC = 12 cm. Find the lengths of AD, BE and CF

**Solution:**

We know that tangent segments to a circle from the same external point are congruent.

Now, we have

AD = AF, BD = BE and CE = CF

Now, AD + BD = 14 cm .....(1)

AF + FC = 12 cm

⇒ AD + FC = 12 cm .....(2)

BE + EC = 8 cm

⇒ BD + FC = 8 cm .....(3)

Adding all these we get

AD + BD + AD + FC + BD + FC = 34

⇒2(AD + BD + FC) = 34

⇒AD + BD + FC = 17 cm .....(4)

Solving (1) and (4), we get

FC = 3 cm

Solving (2) and (4), we get

BD = 5 cm = BE

Solving (3) and (4), we get

and AD = 9 cm