Question

the medians of triangle are 9,12,15 find its area

Answer 1

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In this triangle medians AD =12, BE =9 and CF=15.Because the median cut each other on centroid G in the ratio of 2:1 so the segments AG= 8, GD=4, BG=6, GE=3, CG=10 AND GF= 5
Now we take Triangle ABG, in this Triangle we are seeing that one side AG=8 and other side BG= 6 so the third side AB should be 10 and the triangle ABG is a right angled triangle making an angle 90 degree at the point G.
Now take a bigger triangle ABE. In this triangle we know that AG is perpendicular to BE and so the area of the Triangle ABE should be
1/2 AG*BE= 1/2 * 8*9 = 36
Now we know that any median of triangle bisects the triangle into two triangle of equal areas so the median BE bisects the Triangle ABC into two triangles, ABE and BED, of Equal area so 
area of ABE = area of BED = 36 so the area of triangle ABC= area of ABE+ area of BED = 36+36= 72cm².

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