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Are all angles of triangle ABC smaller than 90 degrees? [#permalink]
 19 Mar 2012, 02:43
Question Stats:
30% (02:07) correct
69% (01:42) wrong based on 1 sessions
Are all angles of triangle ABC smaller than 90 degrees? (1) 2AB = 3BC = 4AC (2) AC^2 + AB^2 > BC^2
Last edited by Bunuel on 19 Mar 2012, 04:32, edited 2 times in total.
Edited the question
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Re: GMAT club Geometry Q [#permalink]
19 Mar 2012, 03:09
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]
19 Mar 2012, 04:56
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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]
19 Mar 2012, 10:04
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]
01 Aug 2012, 08:07
"(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]
01 Aug 2012, 08:14
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]
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]
02 Aug 2012, 22:46
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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ratio of sides in a triangle - relationship with angles? [#permalink]
13 Jan 2013, 09:28
Are all angles of triangle ABC smaller than 90 degrees?
(1) 2AB=3BC=4AC
(2) AC^2 + AB^2 > BC^2
Can we say that if we know the ratio of the sides of a triangle , we can get to know the ratio of the angles of the triangle.
Is the ratio of the sides of a traingle similar to the ratio of the angles of the triangle?
is there any relationship between ratio of sides and its ratio of angles?
Cheers
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Re: ratio of sides in a triangle - relationship with angles? [#permalink]
13 Jan 2013, 09:34
gmatrant wrote: Are all angles of triangle ABC smaller than 90 degrees?
(1) 2AB=3BC=4AC
(2) AC^2 + AB^2 > BC^2
Can we say that if we know the ratio of the sides of a triangle , we can get to know the ratio of the angles of the triangle.
Is the ratio of the sides of a traingle similar to the ratio of the angles of the triangle?
is there any relationship between ratio of sides and its ratio of angles?
Cheers
gmatrant Merging similar topics. Please ask if anything remains unclear. P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to the rule #3: the name of a topic (subject field) MUST be the first 40 characters (~the first two sentences) of the question.
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]
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]
26 Feb 2013, 04:03
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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Re: Are all angles of triangle ABC smaller than 90 degrees?
[#permalink]
26 Feb 2013, 04:03
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