A triangle has side lengths of a, b, and c centimeters. Does each angl

The question asks whether each angle is less than 90 degrees. The only thing most of us know about sides and measure of angles is that in a right triangle, a^2 + b^2 = c^2 where c is the side opposite to 90 degrees angle.

What happens when we have an obtuse triangle? Say the legs stay the same. In the obtuse triangle, the angle will increase and with it the side opposite this angle (c) will increase while a and b stay the same. So we can reason that in an obtuse triangle, a^2 + b^2 < c^2. Then in an acute triangle, a^2 + b^2 > c^2

(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 \(cm^2\), 4 \(cm^2\), and 6 \(cm^2\), respectively.

Area of a semicircle \(= (\pi/2)*r^2 = (\pi/8)*d^2\)

Each side is d, the diameter. The ratio of the areas gives the ratios of the square of the sides. Ignoring the constant since we can cancel them across the equation, we see that 3 + 4 > 6 so it must be an acute triangle. Sufficient.


(2) c < a + b < c + 2

c < a+b holds for all triangles. I am not sure what to do with c + 2. Let me go back to the right triangle and try one with sides 1, 1 and \(\sqrt{2}\) (close to the values being discussed).
\(\sqrt{2} < 1 + 1 < \sqrt{2} + 2\)
Now if c is slightly greater than \(\sqrt{2}\), the inequality will still hold but it will become an obtuse triangle.
Now if c is slightly less than \(\sqrt{2}\), the inequality will still hold but it will become an acute triangle.
Not sufficient

Answer (A)