Data Sufficiency

BTGmoderatorDC wrote:Is triangle ABC obtuse angled?

(1) a^2 + b^2 > c^2

(2) The center of the circle circumscribing the triangle does not lie inside the triangle.

OA E

Source: Official Guide

Question: Is triangle ABC obtuse angled?

Let's take each statement one by one.

(1) a^2 + b^2 > c^2

Case 1: Say an obtuse-angled triangle has sides: a = c = 2 and b = 3. Since for a right-angled triangle, the hypotenuse would be 2√2 and b = 3 > 2√2, the triangle with sides 2, 2, and 3 is an obtuse-angled triangle. We have 2^2 + 3^2 > 3^2 => 4 + 9 > 4 => 13 > 4. The answer is Yes.

Case 2: Say an equailateral-angled triangle has sides: a = b = c = 2. We have 2^2 + 2^2 > 2^2 => 4 + 4 > 4 => 8 > 4. The answer is No.

(2) The center of the circle circumscribing the triangle does not lie inside the triangle.

Case 1: Take an obtuse-angled triangle with an angle 120º and a relatively long base. You will find that the center of the circle circumscribing the triangle does not lie inside the triangle. The answer is Yes.

Case 2: Take a right-angled triangle with a relatively long base. You will find that the center of the circle circumscribing the triangle does not lie inside the triangle. The answer is No.

(1) and (2) together

Even after combining the statements, we can't conclude whether the triangle ABC obtuse angled since both the cases are applicable here too.

The correct answer: E

Hope this helps!

-Jay
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