Are all angles of triangle ABC smaller than 90 degrees?

Question

(1) 2AB = 3BC = 4AC
(2) AC^2 + AB^2 > BC^2

m17 q04

Bunuel · Accepted Answer

We don't need to calculate anything for this question.

Official Solution:

Are all the angles in triangle ABC less than 90 degrees? 
 
(1)  2AB = 3BC = 4AC . 
 
From this information, we have the ratio of the sides:  AB:BC:AC=6:4:3 . All triangles with this ratio are similar and have the same fixed angles. Regardless of the actual values of these angles, the key point is that we can determine them and thus answer the question. Sufficient.
 
(2)  AC^2 + AB^2 > BC^2 .
 
This condition holds for an equilateral triangle (all angles are 60 degrees) as well as for a right triangle with a right angle at either vertex B or C (so in this case, one angle would be 90 degrees; for example, consider a right triangle with  AC=5 ,  AB=4 , and  BC=3 ). Not sufficient.

Answer: A

Hope it's clear.

GyanOne · Answer

Statement 1: 2AB = 3BC = 4AC
=> AB = 2AC and BC = 4/3 AC
=> AB^2 + BC^2 = 4AC^2 + 16/9 AC^2 = 52 AC^2/9 
=> AC^2 > AB^2 + BC^2 
=> Angle B is obtuse
=> All angles are not acute. Sufficient.

Statement 2: AC^2 + AB^2 > BC^2
=> Angle A is acute
But this tells us nothing about the other angles. Insufficient.

Therefore (A) is the answer.

shirisha091 · Answer

Why is the ratio 6:4:3 for AB:BC:AC? Not seeing where the 6 came from.

EvaJager · Answer

Divide through the given equality 2AB = 3BC = 4AC by 12 and get  \frac{AB}{6}=\frac{BC}{4}=\frac{AC}{3} , which can also be written as AB:BC:AC = 6:4:3.

greatps24 · Answer

(2) AC^2 + AB^2 > BC^2 --> this condition will hold for equilateral triangle (all angles are 60 degrees) as well as for a right triangle with right angle at B or C (so in this case one angle will be 90 degrees, for example consider a right triangle: AC=5, AB=4 and BC=3). Not sufficient.

Bunuel,

Can you please explain the above mentioned statement. I am not able to understand

Bunuel,

Can you please explain the above mentioned statement. I am not able to understand

Bunuel · Answer

If AC=AB=BC=1 (satisfies AC^2 + AB^2 > BC^2) --> ABC is an equilateral triangle (all angles are 60 degrees) --> all angles are less than 90 degrees.
If AC=5, AB=4 and BC=3 (satisfies AC^2 + AB^2 > BC^2) --> ABC is a right triangle --> NOT all angles are less than 90 degrees.

Hope it's clear.

WholeLottaLove · Answer

Are all angles of triangle ABC smaller than 90 degrees?

(1) 2AB = 3BC = 4AC
(2) AC^2 + AB^2 > BC^2

A triangle where a^2 + b^2 = c^2 is a right triangle, but is there any way of determining if measures greater than or less than that (say, if a^2 + b^2 was greater than c^2) lead to a triangle having measures greater than or less than 90?

I see how 1 is sufficient. ABC will always have that exact ratio of side lengths. The triangle could be 2x3x4 or 200x300x400 but the angle measurements will always be the same.

Bunuel · Answer

You can see why (2) is not sufficient here: are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html#p1060697 As for your other question, say the lengths of the sides of a triangle are a, b, and c, where the largest side is c. For a right triangle: a^2 +b^2= c^2 . For an acute triangle: a^2 +b^2>c^2 . For an obtuse triangle: a^2 +b^2

WholeLottaLove · Answer

Then wouldn't 2) be sufficient? According to  a^2 +b^2<c^2 , AC^2 + AB^2 > BC^2 shows that the two legs combined are greater than the third implying that this is an acute triangle.

Bunuel · Answer

No, because we don't know whether BC is the largest side.

Ergenekon · Answer

Bunuel, could you explain how we will find angles when we have ratio of sides as in 1). I understand that it will have the same angles, but how can we find them?

Bunuel · Answer

Finding the angles is not our aim (and this is not what you need to know for the GMAT). The aim is to determine whether we CAN find them.

NickHalden · Answer

Bunuel ,

From your analysis, 
a^2 + b^2 < c^2 ------------------ (c is largest then obtuse angle)
so,
3^2 + 4^2 < 6^2 -------------------- Then we have an obtuse angle.

So statement 1 becomes insufficient,
Plz help to clarify....

Bunuel · Answer

The question asks: are all angles of triangle ABC smaller than 90 degrees?

Any triangle which has the ratio of the sides 6:4:3, will be an obtuse triangle, and thus we have an YES answer to the question:

Ananabanana · Answer

Hi there, Sorry I couldn't understand why it can be inferred like this. (Though it doesn't impact to the "sufficiency" of conclusion but I just want to make sure my understand is correct) Isn't AB supposed to be the longest side? (logically because only 2 times AB is already 3 times BC and 4 times AC, so AB is the longest and AC is the shortest side) AB^2 = 4AC^2 BC^2 + AC^2 = 16/9 AC^2 + AC^2 = 25/9* AC^2 As 25/9 = (5/3)^2 <(6/3)^2 = 2^2 = 4, => BC^2 + AC^2 angle C, not B, is an obtuse angle, right? Thanks!

ANSHUMANDASHISB · Answer

So, in this rule - For a right triangle: a2+b2=c2a2+b2=c2. For an acute triangle: a2+b2>c2a2+b2>c2. For an obtuse triangle: a2+b2

Aishyk97 · Answer

Hi, Is B insufficient because we don't know which side is the largest side?

Bunuel · Answer

Yes, if we were given the information that BC is the largest side in triangle ABC and that AC^2 + AB^2 > BC^2, we could deduce that triangle ABC is acute, meaning that all of its angles are less than 90 degrees.

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Are all angles of triangle ABC smaller than 90 degrees?

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