M17 # 04

Can someone please help me here, i am not very sure if i am missing something.

2nd statement of the question, holds true i think for the question to be correct. So it should be D.
However, A is the official answer.

So either the statement 2 needs to be changed to 'greater than equal to' for the OA to hold true. OR the OA needs to be corrected.

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.

Here i think is a general rule

For any triangle with side a, b, c
a^2 = b^2 + c^2 - 2bc Cos(t) where t is the angle between the sides b and c

If t = 90, this is a right angled triangle and the rule becomes the pythagorus theorem.
If t < 90, then cos t > 0, => a^2 < b^2 + C^2

For a triangle to be acute, all three angles must be < 90
So the above rule must be satisfied for all three sides.

In other words, a triangle is acute if
a^2 < b^2 + C^2, b^2 < a^2 + c^2, and c^2 < a^2 + b^2

Condition 1 - does not satisfy the above condition, so we know that all three angles are not < 90

Condition 2 - satisfies one of the condition, but we do not know about the other 2. - Insufficient

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."

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.