SOLUTION: If you have an equilateral triangle and the sides are 10, then what is the altitude?

Draw the altitude (in green, labeled "a") which splits the equilateral triangle
into two congruent right triangles, and splits the bottom 10-unit 
side of the equilateral triangle into two parts, each 5 units long:



Now we look at just the left right triangle:




We use the Pythagorean theorem, where c is the hypotenuse, the
side which is 10 units long, a = the (green) altitude, and b =
the base, or bottom side of the right triangle, which is 5 units
long:

 c² = a² + b²

10² = a² + 5²

100 = a² + 25
  
 75 = a²
 __
V75 = a
 ____
V25·3 = a
  _
5V3 = a

So the altitude "a", is about 8.66 units long.
 
Edwin