We can see that if we split an equilateral triangle in half, we are left with two congruent equilateral triangles. Thus, one of the legs of the triangle is #1/2s#, and the hypotenuse is #s#. We can use the Pythagorean Theorem or the properties of #30˚-60˚-90˚# triangles to determine that the height of the triangle is #sqrt3/2s#.
If we want to determine the area of the entire triangle, we know that #A=1/2bh#. We also know that the base is #s# and the height is #sqrt3/2s#, so we can plug those in to the area equation to see the following for an equilateral triangle:
#A=1/2bh=>1/2(s)(sqrt3/2s)=(s^2sqrt3)/4#
Since #s=10#, #A=(10^2sqrt3)/4=(100sqrt3)/4=25sqrt3" m"^2#
Also, since the height is #sqrt3/2s#, we can say the height is #5sqrt3" m"#.