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Statement (1) by itself is sufficient. S1 completely defines the shape of the triangle, although not its size. Knowing the shape of the triangle, we are able to answer the question.

Statement (2) by itself is insufficient. The inequality holds in an acute-angled triangle (the answer is "yes") but it also holds if angle \(ABC\) is right (the answer is "no"). The correct answer is A.

Regarding statement 1, the angles of a triangle are proportional to the length of the sides. Thus the angles are in the ratio of 2,3 and 4 and are thus all less than 90 degrees.

As for statement 2, note the below: - When the sum of the squares on two sides of a triangle is greater than the square on the third side, the same holds good for other pairs of sides as well, and the triangle is acute . - When the sum of the squares on two sides of a triangle is equal to the square on the third side, then the triangle is right angled . - When the sum of the squares on two sides of a triangle is less than the square on the third side, then the triangle is obtuse angled .

Each statement alone is sufficient to answer the question...

Statment 2 would need to be changed so that side 1 ^2 + side 2 ^2 >= side 3^2

As jakolik mentioned that if the square of both smaller sides is larger than the square of the longest side, then the triangle would be acute, and the statment gives enough info on its own to answer.

I also think that each statement alone is sufficient

2nd statement says us that the sum of squares of 2 sides (side AB and AC) is greater than the third side. This statement implies only and only that the angle contained by AB and AC is acute, i.e angle BAC is acute. Since this angle is acute, one of the other 2 angles can be greater than 90 (as in 30-30-120) or less than 90 (60-60-60) or equal to 90 (30-60-90). Thus this statement is insufficient as we are not sure whether the other angle is greater than or less than 90.

Try these triangles - AB=2, AC=root 3, BC=1 (hint: its a 30-60-90 triangle) and AB=AC=BC=5. Hope it helps.

Ok so in statement 1; a unique ratio tells us that shape is fixed and hence angles are fixed but can we actually determine if it's an acute or an obtuse triangle; I mean do we need to determine the actual angles from the given ratio to prove sufficiency.

Also, using the ratio of the sides as given in statement 1, how can we determine the actual angles?

Regarding statement 1, the angles of a triangle are proportional to the length of the sides. Thus the angles are in the ratio of 2,3 and 4 and are thus all less than 90 degrees.

As for statement 2, note the below: - When the sum of the squares on two sides of a triangle is greater than the square on the third side, the same holds good for other pairs of sides as well, and the triangle is acute . - When the sum of the squares on two sides of a triangle is equal to the square on the third side, then the triangle is right angled . - When the sum of the squares on two sides of a triangle is less than the square on the third side, then the triangle is obtuse angled .

Each statement alone is sufficient to answer the question...

Regards, Jack

Regarding statement 1, "the angles of a triangle are proportional to the length of the sides."

This is definitely not true! There is no such theorem.

The so-called "sinus theorem" states that for any triangle, the following equality holds:

\(\frac{AB}{sinC}=\frac{BC}{sinA}=\frac{AC}{sinB}=2R\), where \(A, B, C\)denote the angles of the triangle and \(R\) is the radius of the circumscribed circle. \(sin\) is the \(sinus\) function, and the value of \(sina\) is not proportional to the value of \(a\).

From statement (1) we can deduce that \(AB = 6x, BC = 4x, AC = 3x\), for certain positive \(x\). Then \(AB^2=36x^2>BC^2+AC^2=25x^2\), which means the triangle is obtuse angled, with angle C greater than 90. Therefore (1) is sufficient.

(2) is not sufficient, as it only proves that angle A is not obtuse, but nothing is known about the other two angles.

Answer A.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.