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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

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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: null - Triangles Examples and Solutions. For the course CBSE
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