Each side of δabc is 75 units in length. point d is the foot of the altitude drawn from side a to side bc. point e is the midpoint of segment ad. what is the length of be

The first thing is to calculate the area of the triangle ABC using the Hero's formula. 
Area = √(s(s-a)(s-b)(s-c))
Where s is half the perimeter and a, b, and c are the lengths of the triangle.
s = 0.5(3×75) = 112.5
Since our triangle is equilateral, a=b=c
Area = √(112.5(112.5-75)³) 
        = √5,932,617.188
        = 2,435.696448 square units.
The same area can be found using the formula, 0.5(bh). Where b is the base length and h is the altitude from the base length.
In this triangle, b=75.

∴ 2,435.696448 = 0.5(75×h)

     h = 4,871.392896÷75
        =64.95190528 units

Since E is the midpoint of AD, then DE=h÷2=32.47595264 units

Now we have a right triangle BDE, where BE is the hypotenuse and BD=75/2. 

∴ BE = √((BD)²+(DE)²)
         =√(37.5²+ 32.47595264²)
         = √2,460.935815
         = 49.6078 units.