A 30-60-90 triangle is drawn on the exterior of an equilateral triangle so the hypotenuse of the right triangle is one side of the equilateral triangle. If the shorter leg of the right triangle is 6 units, what is the area of the equilateral triangle?
If we reflect the 30 60 90 triangle over the y axis, we get an equilateral triangle, congruent to the original one because they share a side. The side length is 12, as 6*2=12.
Now we have to find a formula for the area of an equilateral triangle.
If we draw an median to any side, we will have two congruent triangles, by SSS. Therefore the angles are the same and have to sum to 180, so the angles are 90. Therefore, a median is an altitude and vice versa with altitudes by AAS in an equilateral triangle.
Let the side be x. Then, base is x/2 and height is sqrt (x^2 - x^2/4) = x sqrt 3 / 2. So, area of equilateral triangle = x^2 sqrt 3 / 4.
So, 144/4 = 36 so 36 sqrt 3.