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 ΔABC=54cm2 then find the lengths of sides AB and AC.

Question

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 ΔABC=54cm2 then find the lengths of sides AB and AC.

Filo · Accepted Answer

In the given figure, ΔABC circumscribed the circle with centre O.Radius OD=3cmBD=6cm,DC=9cmArea of ΔABC=54cm2To find :  Length of AB and AC.AF and EA are tangents to the circle at point A.Let AF=EA=xBD and BF are tangents to the circle at point B.BD=BF=6cmCD and CE are tangents to the circle at point C.CD=CE=9cmNow, new sides of the triangle are:AB=AF+FB=x+6cmAC=AE+EC=x+9cmBC=BD+DC=6+9=15cmNow, using Heron's formula:Area of triangle ABC=s(s−a)(s−b)(s−c)​Where S=2a+b+c​S=1/2(x+6+x+9+15)=x+15Area of ABC=(x+15)(x+15−(x+6))(x+15−(x−9))(x+15−15)​ Or54=(x+15)(9)(6)(x)​Squaring both sides, we have542=54x(x+15)x2+15x−54=0Solve this quadratic equation and find the value of x.x2+18x−3x−54=0x(x+18)−3(x+18)=0(x−3)(x+18)=0Either x=3 or x=−18But x cannot be negative.So, x=3Answer :-AB=x+6=3+6=9cmAC=x+9=3+9=12cm

Question Text	In the given figure, a triangle $A BC$ 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 $Δ A BC = 54 c m^{2}$ then find the lengths of sides AB and AC.
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In the given figure, a triangle $A BC$ 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 $Δ A BC = 54 c m^{2}$ then find the lengths of sides AB and AC.

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