Is triangle ABC obtuse angled?

Please see the sketch for clarity.



Edit: Clearer pic updated

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IMO : E

Given ABC is a triangle

Statement 1: \(a^2 + b^2 > c^2\)

Condition for a triangle to be an obtuse is
\(c^2 > a^2 + b^2\) (with "c" largest side)
Since nothing is given about which is the largest side
Not suff

Statement 2 : The center of the circle circumscribing the triangle does not lie inside the triangle.

Circum-center lies on the triangle = Right angled triangle
Circum center lies outside the triangle = Obtuse Triangle
Hence two possible cases, not suff

Combined:
Still not suff

SOLUTION:

Statement 1): a^2 + b^2 > c^2. This condition works in Right angled triangle and Obtuse angled triangle.

Hence , Insufficient.

Statement 2): The Center of the circle circumscribing the triangle does not lie inside the triangle.

Centre of the circle can lie on the triangle or outside the triangle.

Here, Apply relation between Central Angle and Inscribed angle.


if Central Angle makes 180 degrees with any two vertices of the triangle then it is a right angled triangle, in this case centre of the circle is not inside the triangle.

if Central angle makes more than 180 degrees with any two vertices of the triangle then , as per relation between Central angle and Inscribed Angle . the triangle has a angle more than 90 degrees, in this case centre of the circle is not inside the triangle.

Hence , Insufficient.

Both statements together also same results as above.

Hence, Insufficient.

ANS) E

I believe statement 1 is sufficient; I think the correct answer choice is A. I think triangle ABC will always be an acute triangle if statement 1 is true. Can someone offer me a counterexample?

Also I find it hard to believe that this question is from the official guide since the variables: a,b, and c are vague. I think sides AB, BC, and AC should be mentioned or the question should be edited in a way that it is not ambiguous.

Please remember that it is a waste of time to doubt official questions and/or OAs for them. Take them as they are.

Additionally, consider 2 cases.

ABC with sides 3,3,3 in this case, 3^2+3^2>3^2 ---> triangle is NOT an obtuse angled triangle.

But with ABC as 3,4,6 (6^2+4^2>3^2), the angles are 36°, 26°, 118°, you do get a yes for an obtuse angle. Without knowing what do a,b,c, stand for, you wont be able to answer the question.

The ambiguity you are talking about is what makes this statement not sufficient and E as the correct answer.
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Hi Engr2012,

The problem I had with the question was that no where in the question stem does it mention that variables a,b, and c represent the lengths of triangle ABC. So I thought that If I had to assume that a,b,c represent the lengths of a triangle, then I can also assume that length c of triangle of ABC was the biggest as it is when describing the Pythagorean theorem. I agree that the official answer is E because we don't know whether C represents the largest side of the triangle or not.

If you know c is the largest side, then statement will become sufficient.

Hope this helps.
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In your example, the sides of the triangle were 3,4, and 6. If we were to assume that side c was the largest side of the triangle, then your example doesn't satisfy statement 1.
Let's say a=3, b=4, and c=6 then \(a^2 +b^2 <c^2\)
I would expect it to be obtuse triangle.

According to GMAT expert, Mike Mcgarry of Magoosh, the following properties are true if c is the largest side:
If \(a^2+b^2 < c^2\), the triangle is obtuse.
If \(a^2+b^2 > c^2\), the triangle is acute.
If \(a^2+b^2 =c^2\), the triangle is right.

Here is a link to the article: https://magoosh.com/gmat/2012/re-thinkin ... le-obtuse/

I dont think I made myself clear. What I wanted to say with my example above, with sides 3,4,6, until the question tells me that c=largest side, I can play around with the sides to prove that a statement may or may not work. A "sufficient" statement MUST work for ALL different cases possible within scope of the question and the given statement(s).

(3,4,6) does work as an example case for statement 1. You are given a^2+b^2>c^2. No one knows whether c is the greatest side. I can take c=3 or 4 or 6 in this case to prove my point. This ambiguity is what is making this statement not sufficient.

Isnt 6^2+4^2>3^2? Yes. Can you have a triangle with sides 3,4,6 ? Yes.

As per the definition, yes a^2+b^2>c^2 will give you an acute angled triangle only if you know that c is the greatest side. You can not apply this in this question because of the inherent ambiguity.

Hope this clears the confusion.

Additionally, I have updated my post above.
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I completely understand what you are saying. The confusion arose when you originally said,

We both agree that if you know c is the largest side, then statement will become sufficient. Just to clarify, I completely understand that this statement is insufficient without knowing whether c is the largest side or not. But the knowledge of the magnitude of c would certainly be important in determining sufficiency. As we both agree that without knowing whether c is the largest side or not, makes this statement is insufficient.

The correct answer is E.

Hi

If \(a^2+b^2 < c^2\), it means the angle between side a & b is obtuse, the triangle is obtuse.
If \(a^2+b^2 > c^2\), it means the angle between a & b is acute, but since other angle can be obtuse, the triangle cant be taken as acute.
If \(a^2+b^2 =c^2\), it means the angle between a & b is 90 deg, hence the triangle is right.

I hope it helps.



Your interpretation is partially correct. Even if you know c is the largest side, I gave you 2 cases above that make statement 1 as not sufficient.

So in order to answer this particular question, the knowledge of whether side c is the largest is not important.

Hope this helps.[/quote]

In your example, the sides of the triangle were 3,4, and 6. If we were to assume that side c was the largest side of the triangle, then your example doesn't satisfy statement 1.
Let's say a=3, b=4, and c=6 then \(a^2 +b^2 <c^2\)
I would expect it to be obtuse triangle.

According to GMAT expert, Mike Mcgarry of Magoosh, the following properties are true if c is the largest side:
If \(a^2+b^2 < c^2\), the triangle is obtuse.
If \(a^2+b^2 > c^2\), the triangle is acute.
If \(a^2+b^2 =c^2\), the triangle is right.

Here is a link to the article: https://magoosh.com/gmat/2012/re-thinkin ... le-obtuse/[/quote]

I dont think I made myself clear. What I wanted to say with my example above, with sides 3,4,6, until the question tells me that c=largest side, I can play around with the sides to prove that a statement may or may not work. A "sufficient" statement MUST work for ALL different cases possible within scope of the question and the given statement(s).

(3,4,6) does work as an example case for statement 1. You are given a^2+b^2>c^2. No one knows whether c is the greatest side. I can take c=3 or 4 or 6 in this case to prove my point. This ambiguity is what is making this statement not sufficient.

Isnt 6^2+4^2>3^2? Yes. Can you have a triangle with sides 3,4,6 ? Yes.

As per the definition, yes a^2+b^2>c^2 will give you an acute angled triangle only if you know that c is the greatest side. You can not apply this in this question because of the inherent ambiguity.

Hope this clears the confusion.

Additionally, I have updated my post above.[/quote]
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hi

Can anybody please post any image that depicts the scenario perfectly ...?

thanks in advance

If any angle in triangle is grater 90, then that triangle is obtuse.
The question asks whether triangle ABC is obtuse, not whether angle C is greater than 90.

The question is from Official Guide. You can 1 000 times disagree, but it won't help you on GMAT. I'm more than sure that several Math PhD had reviewed this question before GMAT released it. Chanhe your mind and approach OA in official questions as a must. Peace.

If \(a^2+b^2 < c^2\), it means the angle between side a & b is obtuse, the triangle is obtuse.
If \(a^2+b^2 > c^2\), it means the angle between a & b is acute, but since other angle can be obtuse, the triangle cant be taken as acute.
If \(a^2+b^2 =c^2\), it means the angle between a & b is 90 deg, hence the triangle is right.

The requirement that side c to be largest is ONLY required for a^2+b^2 > c^2.
for other cases, c will always be largest. (other inequalities will not be valid if c is not the largest)


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OK I am running through this thread trying to understand what's happening with this:
a) Right triangles: c^2 = a^2 + b^2
b) Acute triangles: a^2 + b^2 > c^2
c) Obtuse triangles: a^2 + b^2 < c^2

SO I have concluded that if WE KNOW:

1. a^2 + b^2 > c^2 and c is the greatest side that it's acute.
2. a^2 + b^2 < c^2 and c is the greatest side that it's obtuse
3. a^2 + b^2 > c^2 MAY OR MAY NOT BE acute if we don't know whether c is the greatest side
4. a^2 + b^2 < c^2 MAY OR MAY NOT BE obtuse if we don't know whether c is the greatest side

Is this right? Even as I write this, I still don't fully understand. Can someone show with examples for each case?
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When, in statement 2, we say "THE CIRCLE cirumscribing the triangle", dont we mean the circle that most perfectly fits around the triangle? The least possible circle?

Im asking because in the drawing by GMATBusters there is a circle that doesnt perfectly fit around the triangle. But I took for granted that the circle was the least possible circle that circumscribes the triangle. And in this case the only possible case where the center of the circle doesnt lie "inside of the triangle" is when it lies "on the hypotenuse of a right triangle"? In all other cases, the least possible circle around a triangle will always have the center inside of the triangle?

Am I correct here?


Edit: I think I get where I went wrong. Circumscribing means that the circle must be in contact with all the vertices of the triangle. And for a very obtuse triangle this circle wont be the "least possible circle surrounding the triangle".

This question is not from the Official Guide. A DS statement in an official question would never present an inequality like "a^2 + b^2 > c^2" without telling you what a, b and c represent, if a, b and c are meant to represent something (lengths, say). As this question is written, Statement 1 could be telling you about the lengths of a triangle, or it could be telling you about the number of apples, bananas, and cantaloupes at a fruit market. I've also never seen the word "obtuse" in an official question, though I suppose it's not impossible it could appear, and the property Statement 2 tests here is not one I'd expect the GMAT to ever test (though again, it's not completely impossible).

This question seems to have been tagged incorrectly on one forum, and then that tag got copied to another forum. You can find the source with a google search though; it's a prep company question. And while it's true that OG questions will all have correct answers, that is not always true of prep company questions, so it is sometimes a good idea to question the 'OA' to some prep company questions. The answer to this question is E, though.

Bambi2021 - when we say one shape 'circumscribes' another, or one shape is 'inscribed' in another, that means the smaller shape 'fits perfectly' inside the larger one. When a triangle or quadrilateral or other polygon is inscribed in a circle (or equivalently when a circle circumscribes one of those shapes) that means the corners of the polygon are on the circumference of the circle. If instead you inscribe a circle in a triangle or square or other polygon, that means the edges of the polygon are tangent to the circumference of the circle, so they touch the circle at just one point.
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THE CIRCLE circumscribing the triangle: it means all the vertices of the triangle lie on the circle.


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