What is the greatest possible area of a triangular region

[GMAT 5/18]

Source: GMAT Prep, mba.com, exam II

I got this question correct by process of elimination as well as drawing it out and making an educated guess. I would like to understand if there is a solid reason (math rule) for this answer. If anyone can help, that would be great.

What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

a. rt3/4
b. 1/2
c. Pi/4
d. 1
e. rt2

I will post the correct answer later (for those of you wishing to try it without seeing the answer first).

Thank you!

[Saurabh Malpani]

GMAT 5/18 I shouldn't have seen this question but I got somehow lured into it because Geo is my fav area!!!

The answer should be A.

Equilateral triangle always has the largest area. Or if you try to solve it by perpendicular triangle way . 30:60:90 triangle always has the largest area!
Mathematical proof can be shown by solving the problem by double integration method, which is out of scope of GMAT so forget about it.
But I think that's the answer.

Saurabh Malpani

Source: GMAT Prep, mba.com, exam II

I got this question correct by process of elimination as well as drawing it out and making an educated guess. I would like to understand if there is a solid reason (math rule) for this answer. If anyone can help, that would be great.

What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?

a. rt3/4
b. 1/2
c. Pi/4
d. 1
e. rt2

I will post the correct answer later (for those of you wishing to try it without seeing the answer first).

Thank you!

[GMAT 5/18]

Saurabh,

Geo is definitely one of my favourite area's also, but sometimes it infuriates me when I get a question wrong....so sometimes its my favourite and sometimes my most hated.

The answer to this question is b.

I guessed correctly when I took the 2nd GMAT Prep exam - I simply drew a circle, and a few different triangles within it (according to the criteria given by the question). The one that "looked" to have the largest area was (as you state) an equilateral triangle (with sides of 1, 1, and rt2), the area of which = 1/2.

I think the difficulty of this question would be pretty high, as it was my 15th question and I had only gotten 2 wrong prior to this question. I am not sure how the GMAT Prep or GMAC works exactly, but I would imagine this to be around a 700 or 700+ type question? Maybe not though - it could just as easily be a 650 question.

Anyways, thanks for your help.

[Saurabh Malpani]

Oh Shit yes you are correct ....I made a silly silly mistake.....

It's cost me heavily in GMAT.

Thanks
Saurabh Malpani

Saurabh,

Geo is definitely one of my favourite area's also, but sometimes it infuriates me when I get a question wrong....so sometimes its my favourite and sometimes my most hated.

The answer to this question is b.

I guessed correctly when I took the 2nd GMAT Prep exam - I simply drew a circle, and a few different triangles within it (according to the criteria given by the question). The one that "looked" to have the largest area was (as you state) an equilateral triangle (with sides of 1, 1, and rt2), the area of which = 1/2.

I think the difficulty of this question would be pretty high, as it was my 15th question and I had only gotten 2 wrong prior to this question. I am not sure how the GMAT Prep or GMAC works exactly, but I would imagine this to be around a 700 or 700+ type question? Maybe not though - it could just as easily be a 650 question.

Anyways, thanks for your help.

[Saurabh Malpani]

So how did you on your Prep 2 exams?

Saurabh Malpani

[GMAT 5/18]

Um, I did alright I guess. Once again, time was a factor. I had to rush through the last 5 questions on the Quant and the last 3 on the Verbal. I really outdid myself on the Verbal (scored a 42; Usually, I score around 38) and performed poorly on the Quant (scored a 44; usually, I score around 48) - Total score: 700.

I guess if I got 700 on the actual exam I wouldn't be dissapointed, but I'd prefer a little higher.

How have you been scoring?

Speaking of silly mistakes, I seem to make 2 (on average) per exam. This time, I worked a problem, found the answer to be E, and clicked on C and clicked confirm! Seriously, it doesn't get worse than that! Just gotta hope the GMAC gods will be kind to be on test day.

[Saurabh Malpani]

I think that's a great score...I am sure you are ready for the day!..
I will be certainly happy to jump over 700, which is the bare minimum requirement for almost all the universities to which I want to apply.

Well ya if you can improve your quant by 3 points or so it will push your score by some 30 points atleast!! that's like 730-A great score in my opinion.

Good luck once again!!!

Saurabh Malpani

Um, I did alright I guess. Once again, time was a factor. I had to rush through the last 5 questions on the Quant and the last 3 on the Verbal. I really outdid myself on the Verbal (scored a 42; Usually, I score around 38) and performed poorly on the Quant (scored a 44; usually, I score around 48) - Total score: 700.

I guess if I got 700 on the actual exam I wouldn't be dissapointed, but I'd prefer a little higher.

How have you been scoring?

Speaking of silly mistakes, I seem to make 2 (on average) per exam. This time, I worked a problem, found the answer to be E, and clicked on C and clicked confirm! Seriously, it doesn't get worse than that! Just gotta hope the GMAC gods will be kind to be on test day.

[Guest]

It's funny how the GMAT can expect you to know certain facts, without assuming that you have the math to prove those things. This problem is a good example - you need to know that the area of an iscoles triangle where the length of the two equal legs is fixed is maximized when it's a 45/45 right triangle. Other problems require you to know that the area of a triangle with a fixed perimeter is maximized when it is an equilateral triangle. There might be some why of proving these things geometrically, but the only way of know of is to take a derivative to find the max/mins and then take the double derivative to figure out if it's a max or min...well outside the scope of the test!

[christiancryan]

One way I think to see it is to fix part of the view and then look at the area formula as you adjust the position of the final leg.

Here's what I mean:

Let's call the triangle CAB. C is the center of the circle, and A and B are on the circle.

Fix CA at 3 o'clock on the clock dial -- horizontal. That will be our base. Imagine a baseline that extends through the base CA horizontally (across to 9 o'clock and out the other side of the circle).

Now, as we sweep CB around the clock like a second hand, we look at the height -- the vertical distance from B to the baseline. Why? Because this is the only remaining "variable" in our area formula: A = 1/2 b h. b is fixed (at 1 unit).

The height is largest when the angle BCA is 90 degrees (at either the very top of the circle or the very bottom). Every other point on the circle is closer to the baseline we've defined.

Does this rise to the level of mathematical proof? Probably not. And it's essentially what you did, GMAT 5/18. But there's something of a methodology in here -- fix in place what you can, and then vary the final parameter and observe the changes to the "outcome" formula (in this case, area of the triangle).

By the way, GMAT 5/18, don't confuse the terms equilateral and isosceles -- the correct triangle is isosceles, not equilateral.[/img]

[imanemekouar]

Can please someone explain the problem since the begining

[agha79]

I am lost on visulizing this as well. is it possible to show the solution with a diagram MGAMT instructors?

[RonPurewal]

I am lost on visulizing this as well. is it possible to show the solution with a diagram MGAMT instructors?

to be most effective, we would probably have to use an animated diagram -- something that is well beyond the capacity of our forum moderators (certainly well beyond my capacity, at least).

did you try following along with christian ryan's demonstration, a couple of posts above? he has given a pretty vivid description of one graphical justification.

the short version:
you know the triangle must have two sides of length 1.
lay one of these sides flat.
to create the largest possible height (which is all you have to do, since the base will be 1), the other side should be vertical.
this creates a 45-45-90 triangle. its area is 1/2(1)(1), or 1/2.

[hi4apoorva]

Huh..
We don't need to visualize much on this one,I knew that equilateral triangle should have maximum area..but this looked like a trick question..and I thought that Isoceles rgt angle is also possible..

Rt3/4 --gives 1.732/4
and 1/2 --gives 2/4

2/4 is obviously greater than 1.732/4...slected B

[RonPurewal]

Huh..
We don't need to visualize much on this one,I knew that equilateral triangle should have maximum area..

hmm?

this is not true; in this problem, the equilateral triangle does not have the maximum area.
go back to the excellent post above by christiancryan, in which he gives a vivid description of this fact.

if you don't want to scroll and find it, here's a link:
post973.html#p973

[vinversa]