Question 251486
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 = (4/3) * sqrt(sm(sm-m1)(sm-m2)(sm-m3))}}}


where:


{{{sm = (1/2) * (m1+m2+m3)}}}. 


see <a href = "http://mathworld.wolfram.com/TriangleMedian.html" target = "_blank">http://mathworld.wolfram.com/TriangleMedian.html</a> for details.


in terms of your problem:


m1 = 4
m2 = 5
m3 = 6
{{{sm = (1/2)*(4+5+6) = (1/2) * (15) = 7.5}}}


area of the triangle is therefore:


{{{A=(4/3) * sqrt((sm) * (sm-m1) * (sm-m2) * (sm-m3))}}}


this becomes:


{{{A=(4/3) * sqrt((7.5) * (7.5-4) * (7.5-5) * (7.5-6))}}}


which becomes:


{{{A=(4/3) * sqrt((7.5) * (3.5) * (2.5) * (1.5))}}}


which becomes:


{{{A=(4/3) * sqrt(98.4375)}}}


which becomes


A = 13.22875656


here's another reference:


<a href = "http://www.math10.com/en/geometry/median.html" target = "_blank">http://www.math10.com/en/geometry/median.html</a>


here's yet another.


<a href = "http://pballew.net/medians.htm" target = "_blank">http://pballew.net/medians.htm</a>