Find the maximum area of triangle DEM We have triangle ABC. On line AB is point M. The line which is paralel with AB cut AC in point D, and CB in E. In which distance "x" we should put line DE to get a maximum area of triangle DEM.

Referring to the diagram below:- !enter image description here

In $\triangle CAB$, let the length of the altitude through $C$ be $1$ unit.

In $\triangle CDE$, let the length of the altitude through $C$ be $x$ unit.

$\triangle CAB \sim \triangle CDE \implies AB : DE = 1 : x$

Area of $\triangle DEM$

$=\frac{1}{2}.DE. (1 – x)$

$=\frac{1}{2}.x. AB. (1 – x)$

Max(Area of $\triangle DEM$)

$= Max \frac{1}{2}.x. AB. (1 – x)$

$= \frac{AB}{2} Max(x – x^2)$ [$AB$ is given as constant]

For the quadratic function $(x – x^2)$, maximum occurs when $x = \frac{-(1)}{2(-1)} = \frac{1}{2}$

That is, $DE$ should be $0.5$ units from $C$ (on the altitude of $\triangle CAB$ through $C$ and its length is $1$ unit).