How to find area of triangle from its medians The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is
a) $48$
b) $144$
c) $24$
d) $72$
I don't want whole solution just give me the hint how can I solve it.Thanks.
 A: Area of a Triangle from the Medians
A triangle is divided in to $6$ equal areas by its medians:
$\hspace{2cm}$
In the case where the two blue triangles share a common side of the triangle, it is pretty simple to see they share a common altitude (dotted) and equal bases; therefore, equal areas.
In the case where the two red triangles share a common $\frac23$ of a median, the altitudes (dotted) are equal since they are corresponding sides to two right triangles with equal hypotenuses and equal vertically opposite angles, and they share a common base; therefore, equal areas.
Now duplicate the original triangle (dark outline) by rotating it one-half a revolution on the middle of one of its sides:
$\hspace{3cm}$
The triangle in green has sides $\frac23a$, $\frac23b$, and $\frac23c$, and by Heron's formula has area
$$
\frac49\sqrt{s(s-a)(s-b)(s-c)}\tag{1}
$$
where $s=(a+b+c)/2$. Thus, each of the $6$ small, equal-area triangles in the original triangle has an area of half of that. Therefore, the area of the original triangle is $3$ times that given in $(1)$:
$$
\frac43\sqrt{s(s-a)(s-b)(s-c)}\tag{2}
$$
A: There exists a formulae giving the area of a triangle in function of its medians. It is
$$A=\frac13\sqrt{2\alpha^2\beta^2+2\beta^2\gamma^2+2\gamma^2\alpha^2-\alpha^4-\beta^4-\gamma^4}$$ where it is clear what are $\alpha,\beta$ and $\gamma$.
A: You know that medians divide a triangle to 6 equal areas. If you find one of them, multiplying with 6 give you the area of whole triangle. Let's denote one area as $S$, now see the figure:

I guess you saw the right triangle.
A: The area of a triangle made by the medians taken as sides is 75% of the triangle of which the medians are given.  Now you can find the area by heron formula and the area thus you get will be 75% of the area of the triangle of which the medians are given. 
A: In this type of questions, given medians always make triplet (a right triangle). From these given triplet area of triangle can be find easily
A=4/3{area of right triangle form by triplet}
As according to your question: 
A=4/3{0.5×(9×12)}
 =72
A: There is a direct formula:
Let
$$s = (m_1+m_2+m_3)/2,$$
Then
$$\text{area} = \frac{4}{3}\sqrt{s(s-m_1)(s-m_2)(s-m_3)}.$$
This gives answer of above question as $72$.
