Question 801311
Draw the triangle.  This is half of an equilateral triangle.  The hypotenuse of the 30-60-90 triangle is two times the short leg.  


Let y=height
Let x=short leg
Already known, {{{7sqrt(2) = h}}}, hypotenuse, same as side of equilateral triangle.


As property of 30-60-90 triangle, {{{h=2x}}}.
According to pythagorean theorem, {{{x^2+y^2=h^2}}}
We already know h, so we want to use the h relationship formula to make equation with only one unknown variable in the pythagorean theorem relationship.
{{{x=h/2}}}, so substitute:
{{{(h/2)^2+y^2=h^2}}}
{{{y^2=h^2-(h/2)^2}}}
{{{y^2=4h^2/4-h^2/4=(1/4)(4h^2-h^2)}}}
{{{y^2=(h^2/4)(3)}}}
{{{y=(h/2)sqrt(3)}}}


Now go back and find {{{x=h/2}}}.  You will find area as {{{(1/2)(h/2)((h/2)sqrt(3))}}}  which is simply {{{(1/2)*base*height}}}.