Bunuel wrote:

The angles in a triangle are \(x\), \(3x\), and \(5x\) degrees. If \(a\), \(b\) and \(c\) are the lengths of the sides opposite to angles \(x\), \(3x\), and \(5x\) respectively, then which of the following must be true?

I. \(c \gt a+b\)

II. \(c:a:b=10:6:2\)

III. \(c^2 \gt a^2+b^2\)

A. I and III only

B. II and III only

C. I only

D. II only

E. III only

we can eliminate right away A and C. in a triangle, the sum of the 2 sides will ALWAYS be greater than the third side.

we are left with B, D, and E.

since we know the angles are in x, 3x, and 5x ratio, the sides must be in the same ratio.

c:a:b=5:1:3 or 10:2:6.

since the order in B is not correct, we can eliminate II, and pick E as the correct answer.

to verify III, suppose a=3, b=10, c=12

c^2= 144

b^2=100

a^2=9

a^2+b^2=109, which is less than c^2. so possible.