Question

A circle is inscribed in a ∆ABC right angled at B as shown in the figure. Find the radius of the circle if BD=1CM, AF=2CM and CE=3CM.​

Answer 1

We need to find the radius of the circle i.e. OD or OF or OE, which is inscribed inside a ΔABC.

• Now,  ar(ABC) = ar(OAB) + ar(OBC) + ar(OAC)

        and area of a triangle = 1/2 × base×height

• So, 1/2 ×AB×BC = 1/2 ×AB×OD + 1/2 ×OE×BC + 1/2 ×OF×AC

         Let OD = OF = OE = r.

• Now, BD = BE because tangents drawn from an external common point to a circle are equal in length. Similarly, AD = AF and CE = CF.

• So, AB = AD+BD = 1+2 = 3cm                                                                                                             BC = BE+CE = 1+3 = 4cm                                                                                             and AC = AF+FC = 2+3 = 5cm

        Now, 12 = 3r + 4r + 5r

         So, r=1cm.

The radius of the circle is 1 cm.

Answer 2

The radius of the circle is 1 cm.

Step-by-step explanation:

We need to find the radius of the circle i.e. OD or OF or OE, which is inscribed inside a ΔABC.

Now,  ar(ABC) = ar(OAB) + ar(OBC) + ar(OAC)

The formula:

Area of a triangle = 1/2 × base×height

So, 1/2 ×AB×BC = 1/2 ×AB×OD + 1/2 ×OE×BC + 1/2 ×OF×AC

Let OD = OF = OE = r.

Now, BD = BE because tangents drawn from an external common point to a circle are equal in length. Similarly, AD = AF and CE = CF.

So, AB = AD+ BD = 1+2 = 3 cm                                                                             BC = BE+CE = 1+3 = 4 cm                                                    

AC = AF+FC = 2+3 = 5 cm

Now, 12 = 3r + 4r + 5r

12 = 12 r

r = 12/12

r = 1 cm.

The radius of the circle is 1 cm.

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