Good one

[kaushikt]

The sides of a triangle is a, b, and c. Are the three angles all less than 90 measure degrees?
1). The areas of the semi-circles with the radius a, b, c are 4, 5, 6, respectively.
2). c<a+b<c+2

[vbhup2]

I think it is A. Doubtful though.

(1) SUFFICIENT.
pi * r^2 / 2 = 4 => r = sqrt (8/pi) = a
pi * r^2 / 2 = 5 => r = sqrt (10/pi) = b
pi * r^2 / 2 = 6 => r = sqrt (12/pi) = c (largest side)

If c^2 >= a^2 + b^2, then one of the angles >=90. (???)
Test: Is 12/pi >= 8/Pi + 10/pi (or 18/pi). NO. Hence, angles are less than 90.

Hence, SUFFICIENT.

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

1:1:sqrt(2) in the form of a:b:c for a 45-45-90 triangle satisfies the condition above. Hence, one angle could be 90. But, 1:2:2 in the form of a:b:c yields a triangle with all angles less than 90. Hence, INSUFFICIENT.

I guess there are better/easier ways to solve this. Anybody?

BTW, What's the Official Answer?

[kaushikt]

Official Answer is A

[vbhup2]

Anybody with a better solution for statement (2) above?