GMAT Question of the Day - Data Sufficiency - Geometry
In the diagram above ABCD is a rectangle. EB is a semi-circle with center O and a radius of 2. What is the area of triangle CBF?
(1) Angle EFD = 45 degrees
(2) Angle BEF = Angle FBE
How do you prove insufficiency in a GMAT geometry question? Either show that two values are possible for the answer or (and this is basically the same idea) show that the shape can be distorted to produce many different values. Sometimes it's tough to show different values because you can't necessarily calculate the different scenarios. In these cases it's easier to show that the shape can be distorted.
(1) You can see in the diagrams that if the height of the rectangle changes we can still maintain the 45 degree angle by shifting point F to left and consequently change the area of triangle CBF. Insufficient.
(2) This means that triangle EBF is isosceles but still the angles can vary so the area of triangle CBF can also vary. Insufficient.
(1) + (2) If you put the statements together you know that EBF is a right isosceles and that CBF is also a right isosceles. The sides are in a ratio of 1:1:root 2. With this information you can calculate all of the sides and have a definite value for the area of triangle CBF. Sufficient.