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If points A, B, and C form a triangle, is angle ABC>90 degre 16 Nov 2007, 08:59
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If points A, B, and C form a triangle, is angle ABC>90 degrees?

(1) AC = AB + BC − 0.001

(2) AC = AB

M15-24

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If points A, B, and C form a triangle, is angle ABC>90 degre 21 Apr 2014, 06:05
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HI Bunnel,

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

Thanks.
Show more

THEORY:

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 (a triangle that has all angles less than 90°) triangle: \(a^2 +b^2>c^2\).
For an obtuse (a triangle that has an angle greater than 90°) triangle: \(a^2 +b^2<c^2\).


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

(1) AC = AB + BC - 0.001.

If AC=0.001, AB=0.001 and BC=0.001, then the triangle will be equilateral, thus each of its angles will be 60 degrees.

If AC=10, AB=5 and BC=5.001, then AC^2>AB^2+BC^2, which means that angle ABC will be more than 90 degrees.

Not sufficient.

(2) AC = AB --> triangle ABC is an isosceles triangle --> angles B and C are equal, which means that angle B cannot be greater than 90 degrees. Sufficient.

Answer: B.

Similar questions to practice:
are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html
if-10-12-and-x-are-sides-of-an-acute-angled-triangle-ho-90462.html

Hope it's clear.
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If points A, B, and C form a triangle, is angle ABC>90 degre 17 Nov 2007, 23:47
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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.
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If points A, B, and C form a triangle, is angle ABC>90 degre 16 Nov 2007, 21:04
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Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.
Show more


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.
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If points A, B, and C form a triangle, is angle ABC>90 degre 16 Nov 2007, 21:47
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bmwhype2
Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.

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.
Show more


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.
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If points A, B, and C form a triangle, is angle ABC>90 degre 16 Nov 2007, 22:18
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Amit05
bmwhype2
Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.

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.
Show more


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 !!
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If points A, B, and C form a triangle, is angle ABC>90 degre 16 Nov 2007, 23:20
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bmwhype2
Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.

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.
Show more


Please see the correction in blue above.

I pick B.
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If points A, B, and C form a triangle, is angle ABC>90 degre 17 Nov 2007, 01:28
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bmwhype2
Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.

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.
Show more



Since AC = AB + BC - .001, what if BC = 0.001? then AC = AB again as in statement 2.
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If points A, B, and C form a triangle, is angle ABC>90 degre 17 Nov 2007, 23:39
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Amit05
bmwhype2
Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.

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.
Show more


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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If points A, B, and C form a triangle, is angle ABC>90 degre 17 Nov 2007, 23:55
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AC=0.001, AB=500, BC=500.001 =>ABC~0
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BC=500.002 instead of 500.001
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If points A, B, and C form a triangle, is angle ABC>90 degre 12 Apr 2013, 17:27
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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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If points A, B, and C form a triangle, is angle ABC>90 degre 19 Apr 2014, 12:21
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HI Bunnel,

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

Thanks.
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If points A, B, and C form a triangle, is angle ABC>90 degre 24 May 2014, 20:41
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If points A, B, and C form a triangle, is angle ABC>90 degrees?

(1) AC=AB+BC−0.001

(2) AC=AB



M15-24 in GMATClub tests - I am wondering whether the OA is incorrect?
Show more


IMO Answer is "B"

Statement-1:
AC = AB+ BC - .001,
If AB, BC are quite big numbers (greater than .01), then angle ABC would be greater than 90 degrees. But if length of AB, BC are in the same range of .001, then angle ABC could be acute angle also.
So statement 1 is not sufficient.

Statement -2:
AC= AB, it means angle ABC = angle ACB, now in any triangle sum all the angles is 180 degree, thus ABC +ACB+BAC = 180 degree. Now as ABC = ACB -> 2ABC + BAC = 180 -> ABC = 90 - BAC/2. Hence angle ABC is always less than 90 degree.

Statement 2 is sufficient
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If points A, B, and C form a triangle, is angle ABC>90 degre 05 Mar 2015, 10:53
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Bunuel, can we also claim that when the angle us obtuse c will be greater than a and b?
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If points A, B, and C form a triangle, is angle ABC>90 degre 05 Mar 2015, 11:28
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Ergenekon
Bunuel, can we also claim that when the angle us obtuse c will be greater than a and b?
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Yes, the greatest side is opposite the greatest angle.
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If points A, B, and C form a triangle, is angle ABC>90 degre 10 May 2017, 12:28
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HI Bunnel,

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

Thanks.

THEORY:

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 (a triangle that has all angles less than 90°) triangle: \(a^2 +b^2>c^2\).
For an obtuse (a triangle that has an angle greater than 90°) triangle: \(a^2 +b^2<c^2\).


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

(1) AC = AB + BC - 0.001.

If AC=0.001, AB=0.001 and BC=0.001, then the triangle will be equilateral, thus each of its angles will be 60 degrees.

If AC=10, AB=5 and BC=5.001, then AC^2>AB^2+BC^2, which means that angle ABC will be more than 90 degrees.

Not sufficient.

(2) AC = AB --> triangle ABC is an isosceles triangle --> angles B and C are equal, which means that angle B cannot be greater than 90 degrees. Sufficient.

Answer: B.

Similar questions to practice:
https://gmatclub.com/forum/are-all-angle ... 29298.html
https://gmatclub.com/forum/if-10-12-and- ... 90462.html

Hope it's clear.
Show more


Hi Bunuel, to find an obtuse angle within the constraints set by 1) I did the following (in bold). Is this approach okay?

1) If AB + BC = 100, then angle ABC will be close to 180. This triangle is allowed because AC<AB+AC. I felt that this triangle allowed easier visualization of the obtuse angle.


And, as you stated if all sides = 0.001, then angle ABC will be 60.

2) Means that the triangle is isosceles and therefore has 2 equal angles.
2x+y=180
2x=180-y

Because y cannot be 0, x must be less than 90. Suff.
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If points A, B, and C form a triangle, is angle ABC>90 degre 07 Feb 2018, 01:38
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If points A, B, and C form a triangle, is angle ABC>90 degrees?

(1) AC = AB + BC − 0.001

(2) AC = AB

M15-24
Show more


hi

I don't know whether this is okay, but I tried this problem this way
statement 1 actually says that

AC < AB + BC

now, if we suppose that AC is the largest side, then it is true for any triangle, regardless of whether the triangle is acute or obtuse
so clearly insufficient

statement 2 says that the triangle is an isosceles triangle with its 2 angels equal

now, since the sum of all the angles of a triangle adds up to 180 degrees, angle B cannot even equal to 90 degrees let alone be more than 90 degrees
hence the statement 2 is clearly sufficient

So the answer is B

thanks
:cool:
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If points A, B, and C form a triangle, is angle ABC>90 degre 27 Mar 2019, 19:10
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Experts can someone please explain, Seems like S1 is sufficient. His explanation seems logical .

Amit05
bmwhype2
Points A, B and C form a triangle. Is ABC > 90 degrees?

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

Please explain your answer.

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.
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If points A, B, and C form a triangle, is angle ABC>90 degre 07 Jan 2024, 03:31
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Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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