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Area of Trapezoid

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by MartyMurray » Mon May 30, 2016 6:40 pm
750+ wrote:Image

If BE || CD, and BC = AB = 3, AE = 4 and CD = 10, what is the area of trapezoid BEDC?

A. 12
B. 18
C. 24
D. 30
E. 48
Since BE and CD are parallel, the angles where BE and CD intersect AC are the same, and the angles where BE and CD intersect AD are the same.

Triangles BAE and CAD also share angle A.

Clearly since the three angles of triangle BAE are the same as the three angles of triangle CAD, BAE and CAD are similar triangles.

The lengths of corresponding sides of similar triangles are all in the same ratio.

So if BC = AB, then AC = 2AB and all the sides of CAD are twice the length of the corresponding sides of BAE.

Since AE = 4, AD = 8.

Since CD = 10, BE = 5

CA = 3 + 3 = 6

AD = 8

CD = 10

So CAD is a 6-8-10 (multiple of a 3-4-5) right triangle.

BA = 3

AD = 4

BD = 5

BAD is a 3-4-5 right triangle.

To find the area of a right triangle, you can simply multiply the lengths of the two perpendicular sides and divide by 2.

Area CAD = (6 x 8)/2 = 24

Area BAE = (3 x 4)/2 = 6

To get the area of the trapezoid, subtract area BAE from area CAD.

24 - 6 = 18

The correct answer is B.
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by OptimusPrep » Mon Jun 06, 2016 7:25 am
750+ wrote:The right answer is B (18). Can someone please post a solution. Thanks
Consider the triangle ABE,
The two sides are 3 and 4. Hence the third side would be 5 (3 - 4 - 5 right triangle)
Area = 1/2 * 3 * 4 = 6

In triangle ABC,
The sides of this triangle are twice the sides of triangle ABE, hence area would be 2^2 times.
Area of ABC = 4*6 = 24

Area od the trapezoid = 24 - 6 = 18

Correct Option: B
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by Matt@VeritasPrep » Tue Jun 07, 2016 11:05 pm
Of course, there's also

Image

So the trapezoid is (3/4) of the triangle, or (3/4) * 24.
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by ceilidh.erickson » Fri Jun 10, 2016 1:24 pm
This question is from Mprep CATs.

Other posters have already given you great solutions, but there's something I want to add:

Just because the question asks for the area of the trapezoid, that doesn't mean that we have to apply the trapezoid area formula. In fact, there's no easy way to find the height of the trapezoid. Most students who try this solution path get stuck.

You should always ask yourself: why did they give me all of the information / all of the figures here? If it were possible to find the area of the trapezoid by itself, there would be no reason to include a triangle on top. So we need to figure out what one shape reveals about the other. In this case, the triangle told us that we have similar 3:4:5 triangles, and the trapezoid is:
Big triangle - small triangle

Often in GMAT geometry, it's easier to solve for what a shape isn't rather than what it is.
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