Dear
phoenix9801
Hi, there. I'm happy to help with these. Incidentally, I like the 9801 = 99^2 in your screenname.
For the first, it is awfully helpful to know that the area of an equilateral triangle of side s is
A = sqrt(3)*(s^2)/4
See this blog for an explanation:
https://magoosh.com/gmat/2012/gmat-math- ... emorizing/
Given that, area of the the whole equilateral is sqrt(3)*(6^2)/4 = 9sqrt(3), so one third of that is 3sqrt(3), answer
C.
For the second, we know the hypotenuse is sqrt(36) = 6, and we know the long leg is sqrt(27) = 3*sqrt(3). With the Pythagorean Theorem, we can figure out, the square of the short leg is 9, so the length of the short leg is 3. This is a right triangle in which the short leg is half the length of the hypotenuse --- that has to be a 30-60-90 triangle. See this blog
https://magoosh.com/gmat/2012/the-gmats- ... triangles/
So the angle opposite the long leg in a 30-60-90 triangle is 60 degrees, answer =
C.
For the third, we need the other special triangle, also discussed at
https://magoosh.com/gmat/2012/the-gmats- ... triangles/
We need the 45-45-90 triangle. Here's a quick approach. Let the side of the square = x. Then, consider right triangle ADC --- by the Pythagorean Theorem,
x^2 + x^2 = 4^2
2*x^2 = 16
x^2 = 8
Well, we need do nothing else. The side of the square, squared, is 8, which means the area = 8. Answer choice =
C.
Does all that make sense?
Mike
