How to find area of triangle from its medians

Question

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.

Answer 1

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.

Answer 2

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 $72$.

Answer 3

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

Answer 4

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 $\gamma$.