Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

B states that AE = 10 and triangle ABE is a right triangle. So it makes it a special case "side-based" right triangle where one of the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5.

Side AE = 10, which means that side AB = 6 and side BE = 8 (ratio 6:8:10 = ratio 3:4:5). Now knowing the sides, you can easily calculate the area which equals 24 < 25.

Given that ABCD is a rectangle, is the area of triangle ABE > 25? (Note: Figure above is not drawn to scale).

Attachment:

Rectangle.PNG [ 2.86 KiB | Viewed 15975 times ]

\(Area=\frac{1}{2}*AB*BE\)

(1) AB = 6 --> clearly insufficient: BE can be 1 or 100.

(2) AE = 10 --> now, you should know one important property: for a given length of the hypotenuse a right triangle has the largest area when it's isosceles, so for our case area of ABE will be maximized when AB=BE. So, let's try what is the largest area of a right isosceles triangle with hypotenuse equal to 10. Finding legs: \(x^2+x^2=10^2\) (where x=AB=BE) --> \(x=\sqrt{50}\) --> \(area_{max}=\frac{1}{2}\sqrt{50}^2=25\). Since it's the maximum area of ABE then the actual area cannot be more than 25. Sufficient.

B states that AE = 10 and triangle ABE is a right triangle. So it makes it a special case "side-based" right triangle where one of the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5.

Side AE = 10, which means that side AB = 6 and side BE = 8 (ratio 6:8:10 = ratio 3:4:5). Now knowing the sides, you can easily calculate the area which equals 24 < 25.

Hi, and welcome to GMAT Club.

Unfortunately your reasoning is nor correct.

You assume with no ground for it that the lengths of the sides are integers. Knowing that hypotenuse equals to 10 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 6:8:10. Or in other words: if \(a^2+b^2=10^2\) DOES NOT mean that \(a=6\) and \(b=8\), certainly this is one of the possibilities but definitely not the only one. In fact \(a^2+b^2=10^2\) has infinitely many solutions for \(a\) and \(b\) and only one of them is \(a=6\) and \(b=8\).

For example: \(a=1\) and \(b=\sqrt{99}\) or \(a=2\) and \(b=\sqrt{96}\) or \(a=4\) and \(b=\sqrt{84}\) ...

You assume with no ground for it that the lengths of the sides are integers. Knowing that hypotenuse equals to 10 DOES NOT mean that the sides of the right triangle necessarily must be in the ratio of Pythagorean triple - 6:8:10. Or in other words: if \(a^2+b^2=10^2\) DOES NOT mean that \(a=6\) and \(b=8\), certainly this is one of the possibilities but definitely not the only one. In fact \(a^2+b^2=10^2\) has infinitely many solutions for \(a\) and \(b\) and only one of them is \(a=6\) and \(b=8\).

For example: \(a=1\) and \(b=\sqrt{99}\) or \(a=2\) and \(b=\sqrt{96}\) or \(a=4\) and \(b=\sqrt{84}\) ...

Hope it's clear.

This is what's so great about the forum. One's faulty assumptions get checked in time. In this case, I had also fallen into the trap of thinking that since hypotenuse is 10 the other sides are 8 and 6. As Bunuel points out, that's clearly the wrong way to think about this.

And knowing the isosceles-right triangle property certainly helps!
_________________

If you like it, Kudo it!

"There is no alternative to hard work. If you don't do it now, you'll probably have to do it later. If you didn't need it now, you probably did it earlier. But there is no escaping it."

Given that ABCD is a rectangle, is the area of triangle ABE > 25? (Note: Figure above is not drawn to scale).

Attachment:

Rectangle.PNG

\(Area=\frac{1}{2}*AB*BE\)

(1) AB = 6 --> clearly insufficient: BE can be 1 or 100. (2) AE = 10 --> now, you should know one important property: the right triangle has the largest area when it's isosceles, so for our case area of ABE will be maximized when AB=BE. So, let's try what is the largest area of a right isosceles triangle with hypotenuse equal to 10. Finding legs: \(x^2+x^2=10^2\) (where x=AB=BE) --> \(x=\sqrt{50}\) --> \(area_{max}=\frac{1}{2}\sqrt{50}^2=25\). Since it's the maximum area of ABE then the actual area can not be more than 25. Sufficient.

Answer: B.

Well an isosceles triangle has maximum area given a hypothenuse. The hypothenuse doesn't seem to be given here, side AB can be as long or as short as you want, thereby making the area larger or smaller than 25.

edit: sorry didnt read the question correctly, i somehow read that BE was given as 10.

Given that ABCD is a rectangle, is the area of triangle ABE > 25? (Note: Figure above is not drawn to scale).

Attachment:

Rectangle.PNG

\(Area=\frac{1}{2}*AB*BE\)

(1) AB = 6 --> clearly insufficient: BE can be 1 or 100. (2) AE = 10 --> now, you should know one important property: the right triangle has the largest area when it's isosceles, so for our case area of ABE will be maximized when AB=BE. So, let's try what is the largest area of a right isosceles triangle with hypotenuse equal to 10. Finding legs: \(x^2+x^2=10^2\) (where x=AB=BE) --> \(x=\sqrt{50}\) --> \(area_{max}=\frac{1}{2}\sqrt{50}^2=25\). Since it's the maximum area of ABE then the actual area can not be more than 25. Sufficient.

Answer: B.

Hey Bunuel,

The property you mentioned only stands true when the hypotenuse is fixed and that is the reason it cannot be applied to the option A. Else, the answer would have been D.

Thought I should clarify for the people reading the post.

Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

Show Tags

01 Jan 2014, 04:52

Quote:

(1) AB = 6 --> clearly insufficient: BE can be 1 or 100. (2) AE = 10 --> now, you should know one important property: the right triangle has the largest area when it's isosceles, so for our case area of ABE will be maximized when AB=BE. So, let's try what is the largest area of a right isosceles triangle with hypotenuse equal to 10. Finding legs: x^2+x^2=10^2 (where x=AB=BE) --> x=\sqrt{50} --> area_{max}=\frac{1}{2}\sqrt{50}^2=25. Since it's the maximum area of ABE then the actual area can not be more than 25. Sufficient.

Hi Bunnel, Why cant the reasoning that a right triangle has greatest area when it is isosceles be applied to the first statement as well. Which says AB = 6, hence assuming BE = 6 we would get the area = 1/2*6*6 = 18 < 25

(1) AB = 6 --> clearly insufficient: BE can be 1 or 100. (2) AE = 10 --> now, you should know one important property: the right triangle has the largest area when it's isosceles, so for our case area of ABE will be maximized when AB=BE. So, let's try what is the largest area of a right isosceles triangle with hypotenuse equal to 10. Finding legs: x^2+x^2=10^2 (where x=AB=BE) --> x=\sqrt{50} --> area_{max}=\frac{1}{2}\sqrt{50}^2=25. Since it's the maximum area of ABE then the actual area can not be more than 25. Sufficient.

Hi Bunnel, Why cant the reasoning that a right triangle has greatest area when it is isosceles be applied to the first statement as well. Which says AB = 6, hence assuming BE = 6 we would get the area = 1/2*6*6 = 18 < 25

The property says: for a given length of the hypotenuse a right triangle has the largest area when it's isosceles. Thus you cannot apply it to the first statement.
_________________

Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

Show Tags

14 Mar 2015, 09:20

Bunuel, have you encountered real gmat questions testing this concept: for a given length of the hypotenuse a right triangle has the largest area when it's isosceles ?
_________________

If my post was helpful, press Kudos. If not, then just press Kudos !!!

This is a rarer concept (from the realm of Multi-Shape Geometry), but the GMAT has been known to test it.

The broader issue is more about comparing squares and rectangles though.

For example, compare the areas of this square and rectangles....

10x10 9x11 8x12

Areas: (10)(10) = 100 (9)(11) = 99 (8)(12) = 96

By increasing one side and decreasing the other by an "equivalent amount", the area decreases.

When it does appear on the GMAT, it's often themed around 'percentage change' in side lengths (re: length is 10% greater, width is 10% less), but the pattern is still the same.

Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

Show Tags

10 May 2016, 17:49

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.
_________________

Happy New Year everyone! Before I get started on this post, and well, restarted on this blog in general, I wanted to mention something. For the past several months...

It’s quickly approaching two years since I last wrote anything on this blog. A lot has happened since then. When I last posted, I had just gotten back from...

Happy 2017! Here is another update, 7 months later. With this pace I might add only one more post before the end of the GSB! However, I promised that...

The words of John O’Donohue ring in my head every time I reflect on the transformative, euphoric, life-changing, demanding, emotional, and great year that 2016 was! The fourth to...