SOLUTION: In a triangle, the lengths of three medians are 4, 5 and 6, find the area of the triangle.

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Question 251486: In a triangle, the lengths of three medians are 4, 5 and 6, find the area of the triangle.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
area of a triangle in terms of the median is expressed as follows:

The area of a triangle can be expressed in terms of the medians by:

A+=+%284%2F3%29+%2A+sqrt%28sm%28sm-m1%29%28sm-m2%29%28sm-m3%29%29

where:

sm+=+%281%2F2%29+%2A+%28m1%2Bm2%2Bm3%29.

see http://mathworld.wolfram.com/TriangleMedian.html for details.

in terms of your problem:

m1 = 4
m2 = 5
m3 = 6
sm+=+%281%2F2%29%2A%284%2B5%2B6%29+=+%281%2F2%29+%2A+%2815%29+=+7.5

area of the triangle is therefore:

A=%284%2F3%29+%2A+sqrt%28%28sm%29+%2A+%28sm-m1%29+%2A+%28sm-m2%29+%2A+%28sm-m3%29%29

this becomes:

A=%284%2F3%29+%2A+sqrt%28%287.5%29+%2A+%287.5-4%29+%2A+%287.5-5%29+%2A+%287.5-6%29%29

which becomes:

A=%284%2F3%29+%2A+sqrt%28%287.5%29+%2A+%283.5%29+%2A+%282.5%29+%2A+%281.5%29%29

which becomes:

A=%284%2F3%29+%2A+sqrt%2898.4375%29

which becomes

A = 13.22875656

here's another reference:

http://www.math10.com/en/geometry/median.html

here's yet another.

http://pballew.net/medians.htm