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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 6921 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} ...

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.

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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"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]
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

Re: Given that ABCD is a rectangle, is the area of triangle ABE> [#permalink]
01 Jan 2014, 05:03

Expert's post

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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