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Re: M20 #07 : TRIANGLE INSIDE A CIRCLE [#permalink]
24 Apr 2014, 12:30
I believe that the right answer is C. We need both statements in order to answer the question with a definite and unique answer.
Here is my thought process: AD/BCE
Statement 1: even though this statement is highly enticing, it only provides information that the three sides AB, BC, and AC satisfy Pythagoras Theorem. However, it is not even to figure out one unique answer. I can guarantee that there is more than one solution just that any triangle whose vertices lie on the circles and side goes through the center will always be right (let us remember the ratio interior angle to arc, 1:2). There would be an infinite range of values for the height and the base. Therefore, statement one is not sufficient. (Remember that since statement 1 was not enough, we eliminate options A and D).
Statement 2: this statement is trying to build on the information we just in statement 1. However, knowing that these statements have to be analyzed separately, I forget I ever read statement 1 and now, this information is clearly insufficient because it will only provide the measure of one of the angles. It does not even give us the measure of one of the sides. Therefore, it is impossible to figure out the area. (Remember that since statement 1 was not enough and neither was statement 2, we eliminate options A, D, and B).
Now, I analyze them together. ABC, given those two statements, is a 30-60-90 triangle with sides in the ratio 1:sqrt(3):2. Since the points A,B, and C all lie on the circle, then the hypotenuse of the triangle must be equal to 2 and the other two sides are now known values. Therefore, using the two statements together we would have enough information to answer the question.