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Q.

In figure, a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively, If $AB = 12 cm, BC = 8 cm$ and $AC = 10 cm$ , then find the lengths of AD,BE and CF.

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a

7 cm, 5 cm and 4 cm.

b

7 cm, 5 cm and 3 cm.

c

7 cm, 8 cm and 3 cm.

d

7 cm, 5 cm and 2 cm.

answer is A.

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Detailed Solution

Given that a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively.
We have to find the lengths of AD, BE and CF.
Question Image

Question Image

The length of the tangents which are drawn from an external point to a circle is equal.
As the length of the tangents which are drawn from an external point to a circle are equal. So

A D = A F, B D = B E

and

C E = C F

Thus, A D + B D = 12 … . (1) A F + F C = 10 … . (2) B D + F C = 8

On adding the equations (1), (2), and (3). So,

A D + B D + A F + F C + B D + F C = 12 + 10 + 8 ⇒ 2 (A D + B D + F C) = 30 ⇒ (A D + B D + F C) = 15

From the equations (1) and (4),

12 + FC = 15 ⇒ FC = 15 − 12 ⇒ F C = 3 c m

From the equations (2) and (4),

10 + B D = 15 ⇒ B D = 15 − 10 ⇒ B D = 2 cm

From the equations (3) and (4),

8 + A D = 15 ⇒ A D = 15 − 8 ⇒ A D = 7 cm

The required values are 7 cm, 5 cm and 3 cm.
Therefore, the correct option is 1.

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