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# Solution - the below given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. - CBSE Class 10 - Mathematics

ConceptTriangles Examples and Solutions

#### Question

the below given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. If the
area of ΔABC is 54 cm2, then find the lengths of sides AB and AC.

#### Solution

Let the given circle touch the sides AB and AC of the triangle at points F and E respectively and let the length of line segment AF be x.
Now, it can be observed that:
BF = BD = 6 cm (tangents from point B)
CE = CD = 9 cm (tangents from point C)
AE = AF = x (tangents from point A)
AB = AF + FB = x + 6
BC = BD + DC = 6 + 9 = 15
CA = CE + EA = 9 + x
2s = AB + BC + CA = x + 6 + 15 + 9 + x = 30 + 2x
s = 15 + x
s – a = 15 + x – 15 = x
s – b = 15 + x – (x + 9) = 6
s – c = 15 + x – (6 + x) = 9

Area of triangle ABC =sqrt(s(s-a)(s-b)(s-c))

54=sqrt((15+x)(x)(6)(9))

54=3sqrt(6(15x+x^2))

18=sqrt(6(15x+x^2))

324=6(15x+x^2)

54=15x+x^2

15x+x^2-54=0

x^2+ 18x -3x -54 0=0

x(x-18) - 3(x - 18)=0
(x - 18)(x - 3)= 0
x = 18 and x = 3
As distance cannot be negative, x = 3

AC = 3 + 9 = 12
AB = AF + FB = 6 + x = 6 + 3 = 9

Is there an error in this question or solution?

#### Reference Material

Solution for question: the below given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. concept: Triangles Examples and Solutions. For the course CBSE
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