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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 19 Mar 2012, 01:43
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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.
Edited the question
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Re: GMAT club Geometry Q [#permalink] New post 19 Mar 2012, 02:09
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] New post 19 Mar 2012, 03:56
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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] 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] 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] 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] New post 02 Aug 2012, 18:04
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, 21:46
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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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink] 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] New post 26 Feb 2013, 03:03
Expert's post
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] New post 10 Dec 2013, 09:42
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, 01:20
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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, 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] New post 11 Dec 2013, 06:39
Expert's post
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] 11 Dec 2013, 06:39
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