Problem Solving

You can solve even if you don't know the 30-60-90 ratios. When you draw a height between two equal sides of any isosceles triangle, and in particular in an equilateral triangle, you are always cutting the base exactly in half. So if the length of one side of the equilateral triangle here is x, then when we draw the height, we are cutting the base into two shorter lines each of length x/2. So each right triangle has sides of lengths x/2, 8 and x. Using Pythagoras we can find x:

(x/2)^2 + 8^2 = x^2
x^2/4 + 64 = x^2
64 = 3x^2/4
x^2 = 4*64/3
x = 2*8/√3 = 16/√3

Since the height is 8, and the base is x, the area is

bh/2 = (16 / √3 ) * 8 / 2 = 64 / √3

Mathematicians don't write fractions with square roots in the denominator, so the answer really ought to be written (multiplying on top and bottom of the fraction we just found by √3 ) as 64√3 / 3 .

DavidG@VeritasPrep wrote:
Therefore, the area of each 30:60:90 triangle will be [8*8/(root3)]/2 = 32 root 3. If each of the 30:60:90 triangles has an area of 32 root 3, the area of the full triangle will be 64 root 3.

David - I think you have a couple of typos at the end here, so you might want to edit your post - instead of "32 root3" and "64 root3", I think you meant to write "32 / root3" and "64 / root3".