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# M17#04 - DS

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21 Jun 2010, 01:53
Are all angles of triangle $$ABC$$ smaller than 90 degrees?

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

I am not convinced the OE. Can someone explain more on this?
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21 Jun 2010, 07:00
ykaiim wrote:
Are all angles of triangle $$ABC$$ smaller than 90 degrees?

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

I am not convinced the OE. Can someone explain more on this?

You could just use substitution:

If AC = 90 then 4AC = 360
3BC = 360 then BC = 120

This wouldn't work because three angles must equal 180

If AB = 90 then 2AB = 180
3BC = 180 and BC = 60
4AC = 180 so AC = 45

90 + 60 + 45 = 195

Answer A is sufficient all are less than 90

2. If BC = 80 then AC + AB = 100

However we don't know if AC + AB = 80 + 20 or AC + AB = 95 +5

Both satisfy the equation but one has an angle greater than 90 and the other doesn't
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21 Jun 2010, 07:30
lagomez,

I think AB, BC and AC represents sides.
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21 Jun 2010, 07:52
ykaiim wrote:
lagomez,

I think AB, BC and AC represents sides.

You are right, still same answer.

2AB = 12
AB = 6

3BC = 12
BC =4

4AC = 12
AC = 3

you have a 3-4-6 triangle so one angle is over 90 because if one angle was 90 the ratio would have been 3-4-5 and less than 90 3-4-and a number less than 5

Part 2.

3-3-3 triangle
vs 3-4-6
6^2 + 3^2 > 4^2 sufficient and one angle greater than 90

9 + 9 > 9 sufficient but all angles less than 90
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22 Jun 2010, 07:22
1)
2AB = 3BC = 4AC -> AB = 6X, BC = 4X, AC = 3X

AB^2 > BC^2 + AC^2, so there is one angle bigger than 90°. The answer fo the question is "YES" and therefore statement 1) is Suff.

[strike]2) The square of one side is smaller then the sum of the squares of the other sides, so there is no angle bigger than 90°. The answer to the question is "NO" and therefore statement 2) is Suff[/strike].

My mistake. In the statement 2) you could have a rectangle, equilateral or obtusangle triangles, thefreore it is Insuff.

Last edited by MBAUncle on 22 Jun 2010, 07:33, edited 5 times in total.
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22 Jun 2010, 07:24
Well, S2 in insufficient.
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Re: M17#04 - DS   [#permalink] 22 Jun 2010, 07:24
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# M17#04 - DS

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