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A triangle $A B C$ is drawn to circumscribe a circle of radius $3 cm$ so that the segments $B D$ and $D C$ into which $B C$ is divided by the point of contact $D$ , are of lengths $6 cm$ and $9 cm$ respectively. If the area of $Δ A B C = 54 cm 2$ then the lengths of sides $A B$ and $A C$ will be,

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9, 12

8, 14

6, 10

9, 13

answer is A.

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Detailed Solution

Given are the following,
Here

P A = P B

and

O B ⊥ B A

B D = 6 c m D C = 9 c m O D = 3 c m

The area of

△ A B C = 54 cm 2

Equating between

B F

and

B D

C D

and

C E

A F

and

A E

B F = 6 cm B D = 6 cm C D = 9 cm C E = 9 cm A F = X cm A E = X cm

To calculate

x

we now apply Heron’s formula,
Area of

Δ A B C = 54 cm 2

S =

semi-perimeter.

S = A B + B C + A C 2

S = X + 6 + X + 9 + 9 + 6 2

S = X + 15

Now applying Heron's formula, we get,

A r e a = S (S − A B) (S − B C) (S − A C)

54 = (X + 15) (X + 15 − X − 6) (X + 15 − 15) (X + 15 − X − 9)

54 = (X + 15) 9 × 6 × X

54= X X + 15

54 = X 2 + 15 X

On simplifying the above equation, we get,

X 2 + 15 X − 54 = 0 X 2 + 18 X − 3 X − 54 = 0 X (X + 18) − 3 (X + 18) = 0 (X − 3) (X + 18) = 0

X = 3 c m

[Length cannot be negative]
From i) we have,

A B = X + 6 A B = 9 c m A C = X + 9 A B = 12 c m

Thus the length of

AB

9 cm

and the length of

AC

12 cm

.
Therefore the correct option is 1.

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