Question

А
12. In the given figure, a triangle ABC
is drawn to circumscribe a circle of
radius 3 cm such that the segments
BD and DC into which BC is divided
by the point of contact D, are of
lengths 6 cm and 9 cm respectively. If
the area of AABC = 54 cm? then find
the lengths of sides AB and AC.
[CBSE 2011, '15]
3 cm
D
C
B
+ 6 cm
9 cm​

Answer 1

Step-by-step explanation:

Let AB be divided by E and AC be divided by F

BD = 6cm. (given)

And, BC = 9cm. (given)

BD = BE = 6cm. (tangents from B)

And, BC = CF = 9cm. (tangents from C)

AE = AF = x. (say.) [tangents from A]

Let centre of circle be O

OD = OE = OF = 3cm. (radius)

Area of ΔABC = 54cm² (given)

area of ΔBOC + area of ΔAOB + area of ΔAOC = 54cm²

1/2*BC*OD + 1/2*AB*OE + 1/2*AC*OF = 54cm²

1/2*15*3 + 1/2(x + 6)*3 + 1/2(x + 9)*3 = 54cm²

3/2(15 + x + 6 + x + 9) = 54cm²

2x + 30 = 36cm²

x + 15 = 18

x = 3cm

AB = x + 6 = 3 + 6 = 9cm

AC = x + 9 = 3 + 9 = 12cm

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