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