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Question

In a right angled triangle, medians drawn from the acute angles make an angle of θ which each other and L is the length of the hypotenuse. Then the area of the triangle is equal to:

A
Δ=L2tanθ6
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B
Δ=L2tanθ3
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C
Δ=L2cotθ3
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D
Δ=L2cotθ6
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Solution

The correct option is B Δ=L2tanθ3




Given: angle ABC =90 as Δ is right rt. angle
triangle and G is centroid of Δ as DC & AE are medians .
As, AE median AE;
Length of median AE ;
AE=122b2+2c2a2
Length DC=122b2+2a2c2
Centroid G divides median in ratio of 1: 2
GC=23DC=132b2+2a2c2
& GE=13AE=162b2+2c2a2
& EC=a2
In ΔGEC ; applying cosine formula
cosθ=GE2+GC2EC22GE.GC
=2b2+2c2a236+2b2+2a2c29a242162b2+2c2a2132b2+2a2c2
=2b2+2c2a2+8b2+8a24c29a24(4c2+a2)(4a2+c2)
[using a2+c2=b2 as angle B=90]
cosθ=2b2(4c2+a2)(4a2+c2)




perpendicular = [(4c2+a2)(4a2+c2)4b4]12
=[17a2c2+4a4+4c44a44c48a2c]12
=[9a2c2]12
=3ac
tanθ=3ac2b2
But tanΔ=12ac
tanθ=3Δb2
Δ=b2tanθ3
Δ=L2tanθ3

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