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If median of triangle are 5cm,6cm,7cm then find the area
let the required triangle be ABC and the medians AD = 5 , BE = 6 , CF = 7
let the centroid be O therefore by 2:1 property of centroid DO = 5/3 , BO = 4 and CO = 14/3
now consider triangle BOC and drop a perpendicular on BC and mark the point on which it falls be G
now BD = x and DC = x , BG = x-y and GD = y
appyling pythagoras theoem
16 - (x-y)2 = (14/3)2 - (x+y)2 and
(5/3)2 - y2 = (14/3)2 - (x+y)2
solve the two equations to get x = (√145)/3 and y = 13/(3√145)
now perpendicular OG can be calculated which will come out as √(384/145)
in triangle BOC base BC =2x = 2(√145)/3 and height OG = √(384/145)
area of triangle BOC = (8√6)/3
therefore the area of triangle ABC = 3 ( area of triangle BOC)
= 8√6
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