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What is the area of trapezoid BEDC?

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What is the area of trapezoid BEDC?

by gmattesttaker2 » Wed May 14, 2014 9:36 pm
Hello,

Can you please tell me how to solve this:

In the diagram given if BE is parallel to CD, and BC = AB = 3, AE = 4 and CD = 10, what is the area of trapezoid BEDC?

OA: 18


I tried to solve as follows:

Area (trapezoid BEDC) = 1/2 x (base1 + base2) x height

base1 = BE
base2 = CD

For height, I can draw a line from B to base CD

So, Area (trapezoid BEDC) = 1/2 x (BE + CD) x BH (where H is a point on line CD)
=> Area (trapezoid BEDC) = 1/2 x (BE + 10) x BH

However, I don't know if this is correct since we have 2 unknowns here.

Can you please assist? Thanks a lot.

Regards,
Sri
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by theCodeToGMAT » Wed May 14, 2014 11:14 pm
Triangle ABE is similar to Triangle ACD

SO, AB/AC = AE/AD ==> 3/6 = 4/AD => AD = 8

Also, AB/AC = BE/CD ==> 3/6 = BE/10 ==> BE = 5

AC = 6, CD = 10, AD = 8 ==> Right angled triangle ==> Area = 1/2 * 8 * 6 = 24

Area of ABE = 1/2 * 4 * 3 = 6

Delta Area = 24 - 6 = 18
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by ceilidh.erickson » Thu May 15, 2014 9:52 am
Whenever you have a geometry problem that asks you for dimensions of a figure that you cannot find, step back and find another way to look at it. Sometimes the best way to find the area of a particular figure is to think of negative space - subtract what you don't want from a larger figure.

You're right - there is no way to find the height of that trapezoid (at least no way that the GMAT would expect you to know). So you should ask yourself - why is that larger triangle there? Shapes within other shapes always exist to show us relationships.

Rahul is right - these are similar triangles. Whenever a line within a triangle is parallel to one of the sides, it creates two similar triangles. We can simply find the area of the large triangle, then subtract the area of the smaller one, and we'll be left with the area of the trapezoid, as in Rahul's explanation.

Another possibility (which we can't see without the answer explanations here) would be to estimate. If you're not told that the figure is NOT to scale, you know that it is roughly to scale, and you could possibly eyeball those measurements: "the height seems to be roughly 2..." etc.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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