The lengths of the medians through acute angles of a right-angled triangle are 3 and 4. Find the area of the triangle.

Explanation

`AD = 3, CE =4`

Using Apollonius theorem for median AD, we have
`c^(2) + b^(2) = 2 ((a^(2))/(4) + 9)`..(i)
Using Apollonius theorem for median CE, we have
`b^(2) + a^(2) = 2 ((c^(2))/(4) + 16)`...(ii)
Also `a^(2) + c^(2) = b^(2)` ....(iii)
Adding (i), (ii) and (iii), we get
`3b^(2) = 2 ((b^(2))/(4) + 25)`
or `b^(2) = 20`
Solving (i) and (ii), we get `c= (4)/(sqrt3) and a = 2 sqrt((11)/(3))`
Hence, Area of triangle `= (1)/(2) ((4)/(sqrt3)) (2sqrt((11)/(3)))`
`= (4)/(3) sqrt11`