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Trapezoid

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by parulmahajan89 » Wed Feb 12, 2014 9:58 pm
In a Triangle abc,BE is // to CD, CD=20,BC=AB=6,AE=8 What is the area of trapezoid BEDC?

1)24
2)48
3)66
4)72
5)144

Not sure which approach to apply here? Should I take area of bigger triangle minus area of smaller triangle?
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by Mike@Magoosh » Thu Feb 13, 2014 4:39 pm
parulmahajan89 wrote:In a Triangle abc,BE is // to CD, CD=20,BC=AB=6,AE=8 What is the area of trapezoid BEDC?

1)24
2)48
3)66
4)72
5)144

Not sure which approach to apply here? Should I take area of bigger triangle minus area of smaller triangle?
Dear parulmahajan89,
I'm happy to respond. :-)

I don't think the problem is well specified. First of all, the direction of line segments is not clear --- for example, should D be on the same side of the triangle as E, or the opposite side??

Furthermore, I made two diagrams that conform to the specifications given in the problem, and there's still enough leeway for them to have very different areas.

I think this problem needs a diagram, and I think more information needs to be specified, before it is a well-defined problem that has a meaningful answer.

Mike :-)
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by Bill@VeritasPrep » Fri Feb 14, 2014 11:20 am
I don't think the original version of this problem specifies triangle ABC, but it does provide you with this diagram.
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by Mike@Magoosh » Fri Feb 14, 2014 8:10 pm
parulmahajan89 wrote:In a Triangle abc,BE is // to CD, CD=20,BC=AB=6,AE=8 What is the area of trapezoid BEDC?

1)24
2)48
3)66
4)72
5)144

Not sure which approach to apply here? Should I take area of bigger triangle minus area of smaller triangle?
Image
Ah! Many thanks to Bill for the clarification. A picture is worth a thousand words. Just a note to parulmahajan89: any diagram that accompanies a GMAT math question inevitably contains information that is not reproduced in the verbal prompt; therefore, presenting the verbal prompt without the picture will inevitably be incomplete.

With the diagram, I see this is actually a brilliantly designed question. The first part of the solution involves the properties of similarity. See:
https://magoosh.com/gmat/2013/gmat-math-similar-shapes/
We know that ABE and ACD are similar triangles, and because AB = 6 and AC = 12, we know they are in a 1-to-2 ratio. That means, since AE = 8, AD = 16, and since CD = 20, BE = 10.

The next step is to realize that triangle ABE is a 6-8-10 right triangle, with a right angle at A, and triangle ACD is a 12-16-20 right triangle (these are 3-4-5 triangles multiplied, respectively, by 2 and 4). See:
https://magoosh.com/gmat/2012/pythagorea ... -the-gmat/

Thus, the area of ABC = 0.5bh = 0.5(6)(8) = 24, and the area of ACD = 0.5(12)(16) = 96, so as you suggest, the area of the trapezoid is simply the difference, 96 - 24 = 72. Answer = (D).

Does all this make sense?
Mike :-)
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by parulmahajan89 » Sun Feb 16, 2014 7:32 pm
Thank you. I am sorry I could not post the diagram before.

Can you just tell by looking at the diagram that triangles are prop?

Also is there any other way to solve this question?

Thanks
parul
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