SOLUTION: the lengths of three medians of a triangle are 9,12 & 15cm. The area of the triangle is

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Question 667067: the lengths of three medians of a triangle are 9,12 & 15cm. The area of the triangle is
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

A median in a triangle is a line segment drawn from an angle to the midpoint of the opposite side. The medians of a triangle all intersect in one point. This point divides each median in the ratio 1%3A+2.
let ABC be a triangle
if CM[1]=15...it is equal to the hight of triangle h, AM%5B2%5D=12, and BM%5B3%5D=9
if intersection point of all three medians is T, and AM%5B2%5D is median then
AT%3ATM%5B2%5D=2%3A1 .......=> AT=%2812%2F3%292=8 and TM%5B2%5D=4

and CT%3ATM%5B1%5D=2%3A1 .......=>CT%3ATM%5B1%5D=%2815%2F3%292=10 andTM%5B1%5D=%2815%2F3%291=5
now we can find AM%5B1%5D which is half of base AB
AM%5B1%5D%5E2=%28AT%29%5E2-%28TM%5B1%5D%29%5E2
AM%5B1%5D%5E2=8%5E2-5%5E2
AM%5B1%5D%5E2=64-25
AM%5B1%5D%5E2=39
AM%5B1%5D=sqrt%2839%29
AM%5B1%5D=6.25...........so, the base AB=6.25%2A2=12.5

then the area will be:
A=%281%2F2%2912.5%2A15
A=6.2.5%2A15
A=93.75