Beat The GMAT

neoreaves · Thu Mar 11, 2010 5:53 am

Are all angles of triangle ABC smaller than 90 degrees?

1) 2AB=3BC=4AC

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

OA : Later

Recommend this topic

kstv · Thu Mar 11, 2010 6:38 am

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

Back to top

bpgen · Thu Mar 11, 2010 8:02 am

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).

Back to top

kstv · Thu Mar 11, 2010 8:32 am

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°.

Back to top

yeahdisk · Thu Mar 11, 2010 8:48 am

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)

Back to top

Beat The GMAT

ABC triangle

Free GMAT Prep Resources

GMAT Prep Discounts

Free Upcoming GMAT Events

Latest GMAT Articles

Latest GMAT Videos

Popular Resources

Course Reviews

Course Locations