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# Are all angles of triangle ABC smaller than 90 degrees?

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Director
Joined: 01 Apr 2008
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Name: Ronak Amin
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Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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22 Sep 2009, 12:09
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85% (hard)

Question Stats:

36% (01:19) correct 64% (01:14) wrong based on 83 sessions

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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html
[Reveal] Spoiler: OA

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Senior Manager
Joined: 23 Jun 2009
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Location: Turkey
Schools: UPenn, UMich, HKS, UCB, Chicago
Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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22 Sep 2009, 14:08
Economist wrote:
Are all angles of triangle $$ABC$$ smaller than 90 degrees?

1. $$2AB = 3BC = 4AC$$
2. $$AC^2 + AB^2 > BC^2$$

1) say AB=6 then BC will be 4 and AC will be 3. This is very similar to 3,4,5 triangle. But the long side is longer than that triangle. So angle ACB is greater than 90.
SUFF.

2) Insuff. say AC=11, AB=6 and BC=8. angle ABC is greater than 90 degrees. But AC=8, AB=8 and BC= 8. All the angles are 60 degrees.

A

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

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01 Feb 2015, 12:12
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Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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02 Feb 2015, 01:35
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Economist wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html
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Re: Are all angles of triangle ABC smaller than 90 degrees?   [#permalink] 02 Feb 2015, 01:35
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