Question

In the adjoining figure, sides BC, CA and AB of ABC
Touch a circle at point D, E and F respectively.
If BD = 4 cm, DC = 3 cm and CA = 8 cm, then the length of
Side AB is:

Answer 1

Given :-

• ∆ABC is a ∆.
• Circle Touches sides BC, CA and AB of ∆ABC
• Touch a circle at point D, E and F respectively.
• BD = 4cm.
• DC = 3cm.
• CA = 8cm.

To Find :-

• Side AB ? .

Solution :-

Refer to Image First.

Tangent Theoram :- States that if we draw two lines from the same point which lies outside a circle, such that both lines are tangent to the circle, then their lengths are the same.

So, with Tangent Theoram we get :-

→ AF = AE

→ DC = CE

→ BD = BF .

so,

→ BD = 4cm. (Given).

→ BF = 4cm. --------------- Equation (1).

And,

→ DC = 3cm. (Given).

So,

→ CE = 3cm.

So,

→ AE = CA - CE

→ AE = 8 - 3

→ AE = 5cm.

So,

→ AF also = 5cm. ------------ Equation (2).

Hence, From Equation (1) & (2), we get,

→ AB = AF + FB

→ AB = 5 + 4

→ AB = 9cm. (Ans).

Answer 2

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Using the theorem : tangents drawn from same point to a circle are equal in length.

In the attached figure

CD = CE = 3 cm

and

BF = BD = 4 cm

given that

AB = AE + CE

AE + CE = 8 cm

AE + 3 cm = 8 cm

AE = (8 - 3 ) cm

AE = 5 cm

Since,

AE and AF are tangents drawn from same external point A

therefore,

AE = AF = 5 cm

As we can see in the figure,

AB = AF + BF

AB = 5 cm + 4 cm

AB = 9 cm.