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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?

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

m17 q04
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Jun 2013, 01:49, edited 3 times in total.
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We don't need to calculate anything for this question.

Are all angles of triangle ABC smaller than 90 degrees?

(1) 2AB = 3BC = 4AC --> we have the ratio of the sides: AB:BC:AC=6:4:3. Now, ALL triangles with this ratio are similar and have the same fixed angles. No matter what these angles actually are, the main point is that we can get them and thus answer the question. Sufficient.

(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.

Answer: A.

Hope it's clear.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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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: GMAT club Geometry Q [#permalink]

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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]

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New post 19 Mar 2012, 09:04
1st statement can be used to calculate the angles, hence A is the answer.
The second statement doesn't give any precise info.

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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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New post 01 Aug 2012, 07:07
"(1) 2AB = 3BC = 4AC --> we have the ratio of the sides: AB:BC:AC=6:4:3. Now, ALL triangles with this ratio are similar and have the same fixed angles. No matter what these angles actually are, the main point is that we can get them and thus answer the question. Sufficient."

In reference to Bunnel's explanation -

How can we get the angles from the ratio of the sides? Will the angles be in the same ratio as sides?? This is not true for 45-45-90 and 30-60-90 triangles, the angles do not have the same ratio as the sides? How would you compute the actual angles given the ratio of sides, in statement 1?

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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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New post 01 Aug 2012, 07:14
teal wrote:
"(1) 2AB = 3BC = 4AC --> we have the ratio of the sides: AB:BC:AC=6:4:3. Now, ALL triangles with this ratio are similar and have the same fixed angles. No matter what these angles actually are, the main point is that we can get them and thus answer the question. Sufficient."

In reference to Bunnel's explanation -

How can we get the angles from the ratio of the sides? Will the angles be in the same ratio as sides?? This is not true for 45-45-90 and 30-60-90 triangles, the angles do not have the same ratio as the sides? How would you compute the actual angles given the ratio of sides, in statement 1?


Please, take a look at my previous post:

m17-97962.html#p1108349
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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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]

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New post 25 Feb 2013, 03:51
(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]

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New post 26 Feb 2013, 03:03
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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Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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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

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]

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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

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]

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New post 11 Dec 2013, 05:20
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]

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New post 11 Dec 2013, 06:39
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: GMAT club Geometry Q [#permalink]

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New post 08 May 2014, 06:08
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.


In the above post I think it should be
AC^2 <AB^2 +BC^2 and not AC^2 > AB^2 + BC^2 as AB^2 + BC^2 = 52 AC^2/9 > AC^2

B is acute

For angle C , taking sides as Shown below

BC= (2/3)AB
AC= (2/4)AB

so BC^2 +AC^2 = (4/9 )AB^2+(1 /4)AB ^2 = (25/36)AB^2
so BC^2 +AC^2 < AB^2 hence C is obtuse

Angle C makes it sufficient not B .

Please do correct , if something still needs attention. Thank you

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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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New post 05 Mar 2015, 10:10
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]

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New post 05 Mar 2015, 10:26
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]

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New post 28 Aug 2015, 13:36
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]

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New post 29 Aug 2015, 01:02
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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