Find the area of triangle $\triangle LMN$.

In triangle $\triangle LMN$. $LO$ is median. Also $LO$ is the bisector of angle. If $LO=3\text{ cm}$ and $LM=5\text{ cm}$. Then find the area of triangle $\triangle LMN$.

Now here we will use similarity between $\Delta LMO$ and $\Delta LON$ triangles but I could not proceed further.

• Hint: $3^2+4^2=5^2$ – Henry Jan 14 '17 at 10:50
• Make a picture of the problem, that will make it easier! – Jan Jan 14 '17 at 11:16
• yes i got it now – starunique2016 Jan 14 '17 at 11:17
• the answer will be 12 cm square.3 will be the height and 8 the base of the triangle – starunique2016 Jan 14 '17 at 11:18
• median=bissector line $\implies$ isosceles triangle. – Jean Marie Jan 14 '17 at 11:33

Since $LO$ is both median and the bisector of $\angle NLM$, $\triangle LMN$ is isosceles and $LO \perp MN$.
So $MO = NO= \sqrt{5^2-3^2\ }=4$ and area $\triangle LMN = 4\times 3 =12 \text{cm}^2$
To see that $\triangle LMN$ is isosceles, consider that $\angle LON = 180°-\angle LOM$. Flip $\triangle LON$ to superimpose $\angle NLO$ and $\angle MLO$. Then the only way to make $|MO|=|NO|$ is to have $\angle LON = \angle LOM =\perp$