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In the trapezoid ABCD above, if AD = EC, then the area of the shaded

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In the trapezoid ABCD above, if AD = EC, then the area of the shaded  [#permalink]

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New post 24 Aug 2018, 02:44
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92% (01:19) correct 8% (00:00) wrong based on 24 sessions

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In the trapezoid ABCD above, if AD = EC, then the area of the shaded region is what fraction of the area of the trapezoid ABCD?


A. 1/3

B. 1/4

C. 1/6

D. 1/9

E. 2/3


Attachment:
image035.gif
image035.gif [ 1.03 KiB | Viewed 313 times ]

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Re: In the trapezoid ABCD above, if AD = EC, then the area of the shaded  [#permalink]

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New post 24 Aug 2018, 03:22
Bunuel wrote:
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In the trapezoid ABCD above, if AD = EC, then the area of the shaded region is what fraction of the area of the trapezoid ABCD?


A. 1/3

B. 1/4

C. 1/6

D. 1/9

E. 2/3


Attachment:
image035.gif


Area of trapezoid=Area of rectangle ABED+Area of triangle DEC=AB*DE+1/2*DE*EC=AB*BE+1/2*AB*BE (Since DE=AB, EC=AD=BE)
Or, Area of trapezoid=3/2 AB*BE

& Area of triangle DEC=1/2*AB*BE=Area of shaded region
Or, Area of shaded region=1/3*(3/2 AB*BE)=1/3*Area of trapezoid

Ans. (A)
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Re: In the trapezoid ABCD above, if AD = EC, then the area of the shaded  [#permalink]

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New post 24 Aug 2018, 03:43
Hi,

You can quickly visualize this and we can clear see that, shaded region takes up 1/3rd of the trapezoid(ABCD). Draw the diagonal AE , you can visualize three triangles(congruent).

If you are not convinced with the visualizations, try giving some values to the sides and see the result.

Lets first understand the shapes, ABCD is a trapezoid – means one pair of sides are parallel. Here in our diagram they have given ABED which is a rectangle – opposite sides are equal.

Here lets take BE = 5 and AB = 4, since given EC = AD, then EC = 5 and AD = 5.

Area of trapezoid ABCD = Area of rectangle ABED + Area of triangle ECD(Shaded region)

Area of trapezoid ABCD = 5 *4 + ½ * 5 * 4
= 20 + 10 = 30.

So, Area of shaded region / area of trapezoid = 10/30 = 1/3

So, the answer is A.

Hope this helps.
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Re: In the trapezoid ABCD above, if AD = EC, then the area of the shaded &nbs [#permalink] 24 Aug 2018, 03:43
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