TITLE: triangle with given 2 medians and 1 side
QUESTION [0 upvotes]: I need help with this exercise I got.
We have a triangle with given medians ma=6, mb=9 and side (without given median on that side) c=6. What is the length of a and b and with what values of ma,mb and c we wouldn't have a solution?
I've came to the solution (using trigonometry) that a is around 6 and b 0,7 but I'm not sure of it.

REPLY [1 votes]: Using these formulas which lab bhattacharjee already pointed out in a comment, you get
$$
2b^2+2c^2-a^2 = 4m_a^2 = 144 \\
2a^2+2c^2-b^2 = 4m_b^2 = 324
$$
Subtract $2c^2=72$ from both sides and you get
\begin{align*}
2b^2-a^2 &= 72 & 2a^2-b^2 &= 252 \\
3a^2 &= 2\cdot 252+72 = 576 & 3b^2 &= 2\cdot72+252 = 396 \\
a^2 &= 192 & b^2 &= 132 \\
a &= \sqrt{192} = 8\sqrt3 \approx 13.8564 &
b &= \sqrt{132} = 2\sqrt{33} \approx 11.4891
\end{align*}
So your triangle looks like this:
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