There is a question in the quant problem section that goes like this:
If C is the center of the circle and YZ equals 4, which of the following values is closest to the area of the shaded region?
Then it shows a picture of the circle with the line YZ going through the middle and making a triangle. You’ll have to see the picture to understand.
In the solution, greg solves it by finding that XY = 2, the base of the triangle is 2, and that one of the angles is 60. But that’s not enough information to find that the other side from X to C is 2, unless you use trig. He has a side angle side triangle. So is this a mistake, or is there something I don’t understand about this solution?
\angle CZX == \angle CXZ = 30 as they’re opposite to equal side! thus, \angle ZCX = 120 (all sides of a triangle add to 180!)and \angle YCX = 180-120 = 60 Now, if u drop a perpendicular from X on YC and name it G , you’ll have a 30-60-90 triangle. thus, sides will be in the ratio 1: \sqrt3: 2 You will get the height of triangle (triangle CXY)as \sqrt 3 Now, area of sector(sector is CYX) is \pi \times r^2 \times \frac{\theta}{360} here, \theta=60 and r=2 or ar. of sector is \frac{2}{3} \pi Now, if we subtract the area of triangle from it then we’ll have that segment area, area of triangle(CXY) = \sqrt3 So, \frac{2}{3} \times \pi - \sqrt3 = .36 Thus, ans is A
