1- 2- 3- should I split the square into two triangles then 90 45 45 degree triangle rule?

Q1: Area of shaded region = 1/3*(Area of lmn) Area of lmn = (3)^0.5/4 *(side)^2 Since side = 6 units, Area of lmn = 9*(3)^0.3 Area of shaded region = (1/3)*(9)*(3)^0.5 = 3*(3)^0.5 [spoiler](C)[/spoiler] is correct.

Q2: Ratio of the sides given = (27/36)^0.5 = (3)^0.5 / 2 In a right angled triangle which has angles of 90, 60 and 30 (in degrees), the ratio hypotenuse:second longest side:shortest side = 2:3^(0.5):1 Since the ratio of the given sides is (3)^0.5 /2, the angle opposite to the side measuring (27)^0.5 must be 60 degrees. [spoiler](C)[/spoiler] is correct.

Q3: AC = 4 in If side of the square is 'a', AC = (2)^0.5 times a (2)^0.5 * a = 4 a = 2(2)^0.5 Area of square = a^2 = 8 square inches [spoiler](C)[/spoiler] is correct.

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