In the adjoining figure, a triangle is drawn to circumscribe a circle of radius $2 c m$ such that the segments $B D$ and $D C$ into which $B C$ is divided by the point of contact $D$ are the length $4 c m$ and $3 c m$ respectively. if area of $△ A B C$ is $21 {c m}^{2}$ , find the lengths of sides $A B$ and $A C$ .

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Solution

Given :
$Δ A B C$ is circumscribed a circle with centre O and radius $2 c m .$
Point D divides $B C$ as
$B D = 4 c m, D C = 3 c m, O D = 2 c m$
Area of $Δ A B C = 21 c m^{2}$
Join $O A, O B, O C, O E$ and $O F .$
From figure:
$B F$ and $B D$ are tangents to the circle.
So, $B F = B D = 4 c m$
$C D$ and $C E$ are tangents to the circle.
So, $C E = C D = 3 c m$
$A F$ and $A E$ are tangents to the circle.
Let say, $A E = A F = x c m$
Now,
Area of $Δ A B C =$ x Perimeter of $Δ A B C \times$ Radius
$21 = 1 / 2 [A B + B C + C A] O D$
$21 = 1 / 2 [4 + 3 + 3 + x + x + 4) \times 2$
$21 = 14 + 2 x$
$x = 3.5$
Therefore,
$A B = A F + F B = 3.5 + 4 = 7.5 c m$
$A C = A E + C E = 3.5 + 3 = 6.5 c m$

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In the adjoining figure, a triangle is drawn to circumscribe a circle of radius 2cm such that the segments BD and DC into which BC is divided by the point of contact D are the length 4cm and 3cm respectively. if area of △ABC is 21cm2, find the lengths of sides AB and AC.