TRICKY QUESTION!!!

Since the angles in a triangle must add to 180°, we can write: 5x + 3x + x = 180
Simplify: 9x = 180
Solve: x = 180/9 = 20

So, 3x = 3(20) = 60
And 5x = 5(20) = 100

So, our diagram looks like this:




At this point, I want you to consider a HYPOTHETICAL situation.
IF it were the case that the TOP angle were 90°...

...then we'd have a RIGHT triangle, in which case the Pythagorean Theorem would tell us that p² + m² = n²
In other words, if the top angle were 90°, then Quantities A and B WOULD be EQUAL

KEY POINT: In order to make the top angle 100°, we must OPEN UP sides p and m by 10°, which will INCREASE the length of side n
So, when the top angle is 100°, we can conclude that p² + m² < n²

Answer: A

Cheers,
Brent
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so if its pythagoras
p^2+m^2=n^2

here say p^2+m^2=n^2

but if we make n greater than 90 say 100
so we need to add some part to p and m as well, to cover up larger N
suppose P and M are increased by 5degree equivalents each. how can we be sure?

9x=180
x=20
so we have 20,60,100 as angle measures.
as one angle is obtuse ,we follow the property that n^2>p^2+m^2
hence answer is
A

We can calculate the angles of the triangle using the angle sum property. These turn out to be 100, 60 and 20 degrees. A triangle with any one side greater than 90 degrees is an obtuse angle triangle. For such a triangle, the square of the side opposite to the obtuse angle is greater than the sum of the squares of the other two sides.
Hence, n^2> p^2+m^2

You can assign different values using the largest side is opposite the largest angle. I get D. It's amazing how I have an uncanny ability to see things that the test makers never even considered, and I argue to the death over it and always win. To date, I have had FOUR, yes FOUR, questions thrown out or substantially modified by ETS. Quite the record, if I don't say so myself. I'm a real out-of-the-box thinker that surpasses the GRE, but I unfortunately get penalized because I don't think LIKE THEM. (This has to go into my applications somewhere!)

The correct answer is A
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couldnot understannd

Just wanted to help you out - imagine the lines p and m are connected by a hinge (like a door and the wall). There is a unique triangle for each angle that the hinge has. Now imagine the hinge when the degrees are 90 degrees and the sides are of length p and m. You should be imagining a triangle that looks like the one in the figure . Now imagine that you took out line n, and you changed the degree between p and m so that it is 100. I want you to think - do you think that the distance between the lines is decreasing or increasing? A nice way to think about it is to think about the distance between the endpoints when the angle is 1 vs the angle is 179 - the distance keeps increasing as you increase the angle! So we don't have to change p and m to have an increase in n (If we did that, I believe that the triangles would just be similar and you wouldn't really be changing the degrees).

Let me know if you have any additional questions!

@Libra, fortunately, you were able to solve for the angles in the problem because they gave them with the notation 5x, x, and 3x.

This was a very tricky problem to solve. I got the general approach of finding the angles and then trying to infer what would happen if it were a 90 degree triangle. However, I made a mistake by reasoning wrongly that n^2 < m^2 + n^2 since I had gotten mixed up with the triangle inequality property. Nice question!

Why wouldn't we use the triangle inequality theorem here, which states that one side of a triangle must be less than the sum of the other two sides? With this reasoning, n must be less than p+m so wouldn't the answer be B?

You're correct to say that n < p + m, but that does not necessarily mean n² < p² + m²

Here's an example: If n = 3, n = 2 and m = 2, then it's true that n < p + m (i.e., 3 < 2 + 2)
However, it's NOT true that n² < p² + m² (we get: 3² < 2² + 2², which is NOT true)

Cheers,
Brent
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Why cant we use the inequality property here , where i can assume values of the three sides .
For e.g. in n = 8 and p and m are 5 and 7 respectively then i get B > A .

If i change the value of n to be 7( in between 2 < n < 12)then i get answer A > B .

How can we come to a conclusion in that case?

So when you take \(n = 7\), it means that \(n = m = 7\) (as you've taken \(m = 7\))

And when the sides are equal, the opposite angles should be the same as well but in the figure, it is not.

Thanks alot for the explanation . Got my silly mistake now