Given length of one side and its median and another median in a triangle. Find area of the triangle

Question

Given the triangle $ABC$. If length of medians $AM=7.5$ and $BN=12$ and side $AC=6$, find the area of the triangle.

NOTE: Solution must not use any special formula or trigonometric functions.

My attempt : $NM$ is half of $AB$ in length, Triangles $ANO$ and $MOB$ have same amount of area.

Answer 1

First, use the lengths to solve for the semiperimeter ( which is half the perimeter ). Then set up Heron's formula (which would be area is equal to the square root of semiperimeter multiplied by semiperimeter minus side a multiplied by semiperimenter minus side b multiplied by semiperimenter minus side c. Here are the formulas you need:

S(semiperimeter)= 1/2 (a+b+c)

A (area)=√S (S-a)(S-b)(S-c)

Answer 2

Let $AM$ and $BN$ meet at G(centroid).

Property of the centroid is that it divides a median in the ratio $2:1$.

Therefore $AG=5$ and $GB=4$.

Therefore $ABG$ is a $5,4,3$ i.e right-angled triangle whose area is $10$.

Therefore the total area is $60$