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# Points A, B and C form a triangle. Is ABC > 90 degrees?

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CEO
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Points A, B and C form a triangle. Is ABC > 90 degrees? [#permalink]  16 Nov 2007, 07:59
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45% (medium)

Question Stats:

38% (02:06) correct 61% (01:06) wrong based on 72 sessions
Points A, B and C form a triangle. Is ABC > 90 degrees?

(1) AC = AB + BC - .001
(2) AC = AB
[Reveal] Spoiler: OA
CEO
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Re: C 15.24 degrees of a triangle [#permalink]  16 Nov 2007, 09:29
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

B. if ac = ab, then abc cannot even be 90.
SVP
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[#permalink]  16 Nov 2007, 18:36
can you expand on the above ?
Director
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Re: C 15.24 degrees of a triangle [#permalink]  16 Nov 2007, 20:04
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

I think the answer is A.

S1 :

AC = AB + BC - .001
AC + .001 = AB + BC
Squaring B.S,
(AC + .001) ^ 2 = (AB + BC )^2

AC^2 + 2AC*.001 = AB^2 + 2AB.BC + BC^2
AC^2 = AB^2 + BC^2 + 2AB.BC + 2AC*.001

By Pythagoras Thm, Angle ABC is 90 if AC^2 = AB^2 + BC^2.
Here AC is bigger than that.. which implies that Angle ABC is > 90. Hence Suff.

PS : (I do not know of any formal rule/theorem that states that if Hypotenuse exceeds more than sum of the squares of the sides that the angle > 90. In fact that triangle is no more a right triangle but I just based this on my intuition. I just took an example of triangle with sides 3,4 and 5. )

St2 :

As AC is the Hypo which should be the longest side of a triangle. But here, the Hypotenuse is equal to a side hence this triangle is not a right triangle but we don't know if angle ABC > 90.. Hence In-suff.
Manager
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Re: C 15.24 degrees of a triangle [#permalink]  16 Nov 2007, 20:47
Amit05 wrote:
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

I think the answer is A.

S1 :

AC = AB + BC - .001
AC + .001 = AB + BC
Squaring B.S,
(AC + .001) ^ 2 = (AB + BC )^2

AC^2 + 2AC*.001 = AB^2 + 2AB.BC + BC^2
AC^2 = AB^2 + BC^2 + 2AB.BC + 2AC*.001

By Pythagoras Thm, Angle ABC is 90 if AC^2 = AB^2 + BC^2.
Here AC is bigger than that.. which implies that Angle ABC is > 90. Hence Suff.

PS : (I do not know of any formal rule/theorem that states that if Hypotenuse exceeds more than sum of the squares of the sides that the angle > 90. In fact that triangle is no more a right triangle but I just based this on my intuition. I just took an example of triangle with sides 3,4 and 5. )

St2 :

As AC is the Hypo which should be the longest side of a triangle. But here, the Hypotenuse is equal to a side hence this triangle is not a right triangle but we don't know if angle ABC > 90.. Hence In-suff.

I find myself inclined to agree with your logic about statement 1.

However, I find statement 2 to be sufficient by itself as well.

If ac=ab, then angle ABC = angle ACB.

Therefore angle ABC cannot be greater than 90.
Director
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Re: C 15.24 degrees of a triangle [#permalink]  16 Nov 2007, 21:18
jbs wrote:
Amit05 wrote:
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

I think the answer is A.

S1 :

AC = AB + BC - .001
AC + .001 = AB + BC
Squaring B.S,
(AC + .001) ^ 2 = (AB + BC )^2

AC^2 + 2AC*.001 = AB^2 + 2AB.BC + BC^2
AC^2 = AB^2 + BC^2 + 2AB.BC + 2AC*.001

By Pythagoras Thm, Angle ABC is 90 if AC^2 = AB^2 + BC^2.
Here AC is bigger than that.. which implies that Angle ABC is > 90. Hence Suff.

PS : (I do not know of any formal rule/theorem that states that if Hypotenuse exceeds more than sum of the squares of the sides that the angle > 90. In fact that triangle is no more a right triangle but I just based this on my intuition. I just took an example of triangle with sides 3,4 and 5. )

St2 :

As AC is the Hypo which should be the longest side of a triangle. But here, the Hypotenuse is equal to a side hence this triangle is not a right triangle but we don't know if angle ABC > 90.. Hence In-suff.

I find myself inclined to agree with your logic about statement 1.

However, I find statement 2 to be sufficient by itself as well.

If ac=ab, then angle ABC = angle ACB.

Therefore angle ABC cannot be greater than 90.

Ooops .. I missed that .. I think these are the traps that are set by GMAC to fool us around..

yes, D it is ..

Good question !!
Director
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Re: C 15.24 degrees of a triangle [#permalink]  16 Nov 2007, 22:20
Amit05 wrote:
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

I think the answer is A.

S1 :

AC = AB + BC - .001
AC + .001 = AB + BC
Squaring B.S,
(AC + .001) ^ 2 = (AB + BC )^2

AC^2 + 2AC*.001 = AB^2 + 2AB.BC + BC^2
AC^2 = AB^2 + BC^2 + 2AB.BC + 2AC*.001
--> AC^2 = AB^2 + BC^2 + 2AB.BC - (2AC*.001 + .001^2)

By Pythagoras Thm, Angle ABC is 90 if AC^2 = AB^2 + BC^2.
Here AC is bigger than that.. which implies that Angle ABC is > 90. Hence Suff.

PS : (I do not know of any formal rule/theorem that states that if Hypotenuse exceeds more than sum of the squares of the sides that the angle > 90. In fact that triangle is no more a right triangle but I just based this on my intuition. I just took an example of triangle with sides 3,4 and 5. )

St2 :

As AC is the Hypo which should be the longest side of a triangle. But here, the Hypotenuse is equal to a side hence this triangle is not a right triangle but we don't know if angle ABC > 90.. Hence In-suff.

Please see the correction in blue above.

I pick B.
CEO
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Re: C 15.24 degrees of a triangle [#permalink]  17 Nov 2007, 00:28
jbs wrote:
Amit05 wrote:
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

I think the answer is A.

S1 :

AC = AB + BC - .001
AC + .001 = AB + BC
Squaring B.S,
(AC + .001) ^ 2 = (AB + BC )^2

AC^2 + 2AC*.001 = AB^2 + 2AB.BC + BC^2
AC^2 = AB^2 + BC^2 + 2AB.BC + 2AC*.001

By Pythagoras Thm, Angle ABC is 90 if AC^2 = AB^2 + BC^2.
Here AC is bigger than that.. which implies that Angle ABC is > 90. Hence Suff.

PS : (I do not know of any formal rule/theorem that states that if Hypotenuse exceeds more than sum of the squares of the sides that the angle > 90. In fact that triangle is no more a right triangle but I just based this on my intuition. I just took an example of triangle with sides 3,4 and 5. )

St2 :

As AC is the Hypo which should be the longest side of a triangle. But here, the Hypotenuse is equal to a side hence this triangle is not a right triangle but we don't know if angle ABC > 90.. Hence In-suff.

I find myself inclined to agree with your logic about statement 1.

However, I find statement 2 to be sufficient by itself as well.

If ac=ab, then angle ABC = angle ACB.

Therefore angle ABC cannot be greater than 90.

Since AC = AB + BC - .001, what if BC = 0.001? then AC = AB again as in statement 2.
CEO
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Re: C 15.24 degrees of a triangle [#permalink]  17 Nov 2007, 22:39
GMAT TIGER wrote:
jbs wrote:
Amit05 wrote:
bmwhype2 wrote:
Points A, B and C form a triangle. Is ABC > 90 degrees?

1. AC = AB + BC - .001
2. AC = AB

I think the answer is A.

S1 :

AC = AB + BC - .001
AC + .001 = AB + BC
Squaring B.S,
(AC + .001) ^ 2 = (AB + BC )^2

AC^2 + 2AC*.001 = AB^2 + 2AB.BC + BC^2
AC^2 = AB^2 + BC^2 + 2AB.BC + 2AC*.001

By Pythagoras Thm, Angle ABC is 90 if AC^2 = AB^2 + BC^2.
Here AC is bigger than that.. which implies that Angle ABC is > 90. Hence Suff.

PS : (I do not know of any formal rule/theorem that states that if Hypotenuse exceeds more than sum of the squares of the sides that the angle > 90. In fact that triangle is no more a right triangle but I just based this on my intuition. I just took an example of triangle with sides 3,4 and 5. )

St2 :

As AC is the Hypo which should be the longest side of a triangle. But here, the Hypotenuse is equal to a side hence this triangle is not a right triangle but we don't know if angle ABC > 90.. Hence In-suff.

I find myself inclined to agree with your logic about statement 1.

However, I find statement 2 to be sufficient by itself as well.

If ac=ab, then angle ABC = angle ACB.

Therefore angle ABC cannot be greater than 90.

Since AC = AB + BC - .001, what if BC = 0.001? then AC = AB again as in statement 2.

when dealing with triangles, i usually look for defined size and shape.

-.001 is a concrete size. however, we dont know whether that is a material size that can change the size of the sides of a triangle. From 1, we cannot infer anything.
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[#permalink]  17 Nov 2007, 22:47
1
KUDOS
Expert's post
1. AC=AB+BC-0.001

this is the same as AC>AB+BC (common for triangles)
for example,
AC=1000.001, AB=500, BC=500 => ABC~180

AC=0.001, AB=500, BC=500.001 =>ABC~0

insuf.

2. AB=AC

ABC=ACB => 2ABC<180> ABC<90

suf.

B is correct

P.S if one can draw it solution will come easy.
CEO
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[#permalink]  17 Nov 2007, 22:55
Expert's post
walker wrote:
AC=0.001, AB=500, BC=500.001 =>ABC~0

BC=500.002 instead of 500.001
Manager
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Re: [#permalink]  09 Apr 2013, 19:55
walker wrote:
walker wrote:
AC=0.001, AB=500, BC=500.001 =>ABC~0

BC=500.002 instead of 500.001

AC= AB+BC-0.001

IF AB=100 BC=100.002 & APPLYING IN THE ABOVE FORM IT WOULD GIVE AS

200.001=100+100.002-0.001

I am confused how AC would be equal to 0.001 .. please help me here ..
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Re: Re: [#permalink]  10 Apr 2013, 00:22
Expert's post
FTG2OV wrote:
walker wrote:
walker wrote:
AC=0.001, AB=500, BC=500.001 =>ABC~0

BC=500.002 instead of 500.001

AC= AB+BC-0.001

IF AB=100 BC=100.002 & APPLYING IN THE ABOVE FORM IT WOULD GIVE AS

200.001=100+100.002-0.001

I am confused how AC would be equal to 0.001 .. please help me here ..

Can you please elaborate what you mean?
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Re: Re: [#permalink]  12 Apr 2013, 16:27
If Angle ABC is > 90, then AC has to be the hypotenuse.

With Point 1:

If AB is 1, and BC is 1, then AC would be 1.999, making it the hypotenuse

But if AB is .0006, and BC is .0007, then AC would be .0003, making it not the hypotenuse.

Because the .001 gives us no reference, we cannot conclude anything from Point 1 alone.

If AB = AC, then that means that there is no possible way that AC could be the hypotenuse since there is another side of equal length right next to it. Even if BC is infinitely small, it is still >0 and therefore ABC cannot be >90. Therefore, Point 2 is enough for us to disqualify it alone.
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Re: Re: [#permalink]  19 Apr 2014, 11:21
HI Bunnel,

Could you please provide your comments on statement defined in question.

Thanks.
Re: Re:   [#permalink] 19 Apr 2014, 11:21
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# Points A, B and C form a triangle. Is ABC > 90 degrees?

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