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

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Manager
Joined: 22 May 2007
Posts: 218
Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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02 Aug 2008, 19:59
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Difficulty:

85% (hard)

Question Stats:

44% (02:08) correct 56% (01:30) wrong based on 77 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

m17 q04

OPEN DISCUSSION OF THIS QUESTION IS HERE: are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html
[Reveal] Spoiler: OA
Director
Joined: 27 May 2008
Posts: 541

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02 Aug 2008, 20:27
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aaron22197 wrote:
Are all angles of triangle ABC smaller than 90 degrees?

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

statement 1 : 2AB = 3BC = 4AC
--> AB/6 = BC/4 = AC/3 = k (say)
AB = 6k
BC = 4k
AC = 3k

we can deteremine the shape of the triangle ..... and hence can answer the question

Statement 2: tells that angle A is less than 90, but no information about angle B and C ... so insuff
Intern
Joined: 15 May 2013
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24 Jun 2013, 01:15
from statement 1 we can deduce that AB:BC:AC is 6:4:3

if we compare right angle triange 3,4,5 with above triangle, 3 and 4 are common and 6 is greater than 5. hence in the above triangle one agle has to be greater than 90.

from statement 2 we cannot deduce anything.

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Joined: 02 Sep 2009
Posts: 39697
Re: Are all angles of triangle ABC smaller than 90 degrees? [#permalink]

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25 Jun 2013, 02:51
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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] 25 Jun 2013, 02:51
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# Are all angles of triangle ABC smaller than 90 degrees?

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