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Are all angles of triangle ABC smaller than 90 degrees?

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Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   Updated on: 25 Jun 2013, 02:49
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Are all angles of triangle ABC smaller than 90 degrees?

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

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Originally posted by GMATPASSION on 19 Mar 2012, 02:43.
Last edited by Bunuel on 25 Jun 2013, 02:49, edited 3 times in total.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   19 Mar 2012, 04:56
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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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   19 Mar 2012, 03:09
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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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   02 Aug 2012, 19:04
Why is the ratio 6:4:3 for AB:BC:AC? Not seeing where the 6 came from.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   02 Aug 2012, 22:46
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shirisha091 wrote:
Why is the ratio 6:4:3 for AB:BC:AC? Not seeing where the 6 came from.


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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   25 Feb 2013, 04:51
(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
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   26 Feb 2013, 04:03
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greatps24 wrote:
(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


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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   10 Dec 2013, 10:42
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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   11 Dec 2013, 02:20
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WholeLottaLove wrote:
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.


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: \(a^2 +b^2<c^2\).
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   11 Dec 2013, 06:20
Bunuel wrote:
WholeLottaLove wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

For a right triangle: \(a^2 +b^2= c^2\).
For an acute triangle: \(a^2 +b^2>c^2\).
For an obtuse triangle: \(a^2 +b^2<c^2\).


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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   11 Dec 2013, 07:39
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WholeLottaLove wrote:
Bunuel wrote:
WholeLottaLove wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

For a right triangle: \(a^2 +b^2= c^2\).
For an acute triangle: \(a^2 +b^2>c^2\).
For an obtuse triangle: \(a^2 +b^2<c^2\).


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.


No, because we don't know whether BC is the largest side.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   05 Mar 2015, 11:10
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?
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   05 Mar 2015, 11:26
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Ergenekon wrote:
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?


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.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   28 Aug 2015, 14:36
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....
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   29 Aug 2015, 02:02
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NickHalden wrote:
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....


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:
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   15 Feb 2022, 08:58
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GyanOne wrote:
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.


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 <AB^2
=> angle C, not B, is an obtuse angle, right?

Thanks!
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   08 Aug 2022, 00:06
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<c2a2+b2<c2

A<B<C ? Please let me know. Thanks
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   09 Apr 2023, 06:31
Bunuel wrote:
WholeLottaLove wrote:
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.


You can see why (2) is not sufficient here: https://gmatclub.com/forum/are-all-angl ... l#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: \(a^2 +b^2<c^2\).


Hi, Is B insufficient because we don't know which side is the largest side?
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   09 Apr 2023, 09:45
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Aishyk97 wrote:
Bunuel wrote:
WholeLottaLove wrote:
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.


You can see why (2) is not sufficient here: https://gmatclub.com/forum/are-all-angl ... l#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: \(a^2 +b^2<c^2\).


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


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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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] New post   02 May 2023, 09:37
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]
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