007saisurya
In the diagram above, the four triangles ABE, CBE, ADE, and CDE are all equal, and CD = 5. What is the area between the two circles?
Statement #1: AE = 3
Statement #2: angle BEC = 90 degreesIf the angle is 90 degree in statement 2. Isnt it obvious that the other two sides would be 3 & 4 as the 3,4,5 are pythagorean triplets.
BunuelTwo points:
1. It can be inferred that angle BEC equals 90 degrees from the stem of the problem itself. In triangles ABE, CBE, ADE, and CDE, all angles are equal, and since the sum of the angles' measures at vertex E is 360 degrees, each angle at E must be 90 degrees. Therefore, when you assert that statement (2) is sufficient, you are actually suggesting that the problem's stem itself provides enough information to answer the question, implying that the question is flawed.
2. Just because CD, the hypotenuse in triangle CDE, has a length of 5, it doesn't necessarily mean that the sides of CDE form a Pythagorean triplet. We are not given that the side lengths must be integers. In other words, EC^2 + ED^2 = 5^2 has infinitely many solutions for EC and ED, with only one of them being EC = 3 and ED = 4. For instance, consider EC = 1 and ED = √24; EC = 2 and ED = √21; EC = 1/2 and ED = √24.75, and so on. If we were given that the lengths of EC and ED are integers, then a hypotenuse of 5 would imply that EC and ED are 3 and 4. Otherwise, knowing the length of the hypotenuse alone is insufficient to determine the other two sides.
Hope it's clear.