ABC triangle

[neoreaves]

Are all angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC

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


OA : Later

[kstv]

I) AC =3x, BC = 4x, AB = 6x. 3x²+4x²=5x²
AC² + BC² < AB² if it was = AB² then angle C =90°
As the ratio are given it is possible to say angle C is > 90°
Sufficient.

II) AC²+AB²> BC²
not sufficient as the ratio are not given. Insufficient


IMO A

[bpgen]

Thanks all! Answer would be A alone sufficient. Here is bit detail explanation:

As we know cosine formulas, and cosX always >0 if it's angle <90 degree.
cosA=(-a^2+b^2+c^2)/(2bc)
cosB=(a^2-b^2+c^2)/(2ac)
cosC=(a^2+b^2-c^2)/(2ab)

i.e for acute angled triangle, (-a^2+b^2+c^2) ,(a^2-b^2+c^2) and (a^2+b^2-c^2) should be >0

Therefore, if you take ratio of sides: AC =3x, BC = 4x, AB = 6x
we have:
9+16>36
9+36>16
36+16>9

Hence triangle is acute angled triangle(all angles of triangle ABC smaller than 90 degrees).

[kstv]

In GMAT we try to reason just with the basics.
If one knows the cosines formula very good for him/her as it will help them to solve itquickly .
But I was just thinkin of it is this fashion so as not to feel discouraged if I am unable to recall the formula
I) AC =3x, BC = 4x, AB = 6x.
3x²+4x²=5x²
when AC and BC are 3 and 4 respectively and AB = 5 angle ACB = 90°
the slightest enlongation of AC will make AC and BC diverge or angle ACB will be little > 90°
conversely contraction of AC will make AC and BC converge or angle ACB will be little < 90°
Even at this stage Option A is fine but

angle ACB will be the largest angle as it is opposite the longest side.
It value of ACB whether it is = < or > 90° will surely help decide the other.

Still for more clarity since AB = 6 angle ACB > 90°
angle CAB + ABC < 90°. Individually they have to less than 90°.

[yeahdisk]

I think both are sufficient

the first one is obvious

the second one:

If angle BAC = 90 then AC^2 + AB^2 = BC^2

The only way to change the ratio, without changing the lengths, is to modify the angle BAC...

If you increase the angle above 90, then AC^2 + AB^2 < BC^2
If you decrease the angle below 90, then AC^2 + AB^2 > BC^2

Hence the info is sufficient (all angles are <90)