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# Given that ABCD is a rectangle, is the area of triangle ABE>

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

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04 Feb 2012, 17:12
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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 19382 times ]

(1) AB = 6
(2) AE = 10

How come the answer is B and not C? Can someone please explain?

PS: I tried the jpeg and bitmap format to attach the picture, but it says these two formats are not supported. Therefore attached the .pdf.
[Reveal] Spoiler: OA

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04 Feb 2012, 17:38
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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 19329 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.

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04 Feb 2012, 17:41
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nglekel wrote:
Hi,

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 $$b=\sqrt{84}$$ ...

Hope it's clear.
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Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

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14 Mar 2015, 12:18
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Hi Ergenekon,

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

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.

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

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06 Dec 2013, 11:04
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enigma123 wrote:
Given that ABCD is a rectangle, is the area of triangle ABE > 25?
(Note: Figure above is not drawn to scale).
Attachment:
The attachment Rectangle.PNG is no longer available

(1) AB = 6
(2) AE = 10

How come the answer is B and not C? Can someone please explain?

PS: I tried the jpeg and bitmap format to attach the picture, but it says these two formats are not supported. Therefore attached the .pdf.

F.S 1 is clearly Insufficient.

Another approach for F.S 2 :

We know that $$a^2+c^2 = 10^2 \to a^2+c^2 = 100$$

Also, area of $$\triangle$$ ABE - $$\frac{1}{2}*a*c$$

Is$$\frac{1}{2}*a*c>25 \to$$ Is $$a*c>50 \to 2*a*c>100?$$

Is $$2*a*c>a^2+c^2 \to$$ Is $$(a-c)^2<0$$. Of-course, the answer is NO.Sufficient.
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img.png [ 3.58 KiB | Viewed 15939 times ]

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04 Feb 2012, 17:31
Hi,

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.
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10 Mar 2012, 00:33
Bunuel wrote:
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!
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21 Nov 2012, 07:55
Bunuel wrote:
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.

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.
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19 Oct 2013, 05:44
Bunuel wrote:
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.

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

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06 Dec 2013, 10:05
My first thought was to try and determine the possible lengths of side AB and AE. Is this possible with only one given side measurement?
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Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

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01 Jan 2014, 05:52
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
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Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

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01 Jan 2014, 06:03
Rohan_Kanungo wrote:
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

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

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14 Mar 2015, 10:20
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 ?
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Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

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

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23 May 2017, 22:30
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).

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

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24 May 2017, 00:42
For any right angled triangle the hypotenuse is the largest side and not only isosceles right?

Owing to the convention that the longest side is against the largest angle ?

Posted from my mobile device
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Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]

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24 May 2017, 00:48
SOUMYAJIT_ wrote:
For any right angled triangle the hypotenuse is the largest side and not only isosceles right?

Owing to the convention that the longest side is against the largest angle ?

Posted from my mobile device

Yes. The largest side is always against the largest angle. So, in any right triangle hypotenuse is the largest side.
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Re: Given that ABCD is a rectangle, is the area of triangle ABE>   [#permalink] 24 May 2017, 00:48
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