Vemuri wrote:

GMATBLACKBELT wrote:

Are all the angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC

2)AC^2+AB^2>BC^2

This is an interesting question.

Statement 1 tells us that the sides are in the ratio of 2:3:4. This is sufficient to tell us that the angles are less than 90 degress. How? Simple, use the ratio concept:

2/9*180=40

3/9*180=60

4/9*180=80

Statement 2 tells us that the squares of 2 sides is greater than the square of the 3rd side. This indicates that there is a side which causes an angle greater than 90 degrees, while the other 2 are less than 90. Since the question is asking if all the angles are less than 90, this statement is insufficient to answer the question.

So, the answer is A.

The answer is A, but the reasons are a little different.

I. You can't assume the angles of a triangle are in the same ratio as the sides. Sometimes that's the case (equilateral triangle), but sometimes not (30-60-90 triangle).

FIrst, let's find the ratio of the sides to one another:

2AB = 3 BC, so AB : BC = 3 : 2. 3BC = 4AC, so BC : AC = 4 : 3. To put them all together, multiply AB : BC by 2:

AB : BC : AC = 6 : 4 : 3. If the ratio were 5 : 4 : 3, we'd know that C was a right angle. Since AB is always proportionally longer, we know that C must always be > 90 degrees. SUFFICIENT.

II. Notice that, as the question is worded, there's no guarantee that BC is the longest side of the triangle. So consider these two examples:

a. ABC is equilateral: AC^2 + AB^2 > BC^2.

b. ABC is a right triangle, with C the right angle (so that AB is the hypoteneuse): AC^2 + AB^2 > BC^2.

Each of these cases is consistent with statement II, but they yield different answers for the question, are all angles < 90. So, INSUFFICIENT.