Which is the height of the parallelogram and the triangle?

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sat Nov 23, 2013 11:03 am

Hello,

For this problem:

In parallelogram ABCD, point E lies on side AD. If the area of the parallelogram
is three times the area of triangle ABE,what is the ratio of the length of AE to
the length of AD?

If I draw the following parallelogram (shown in the attachment) would AB be the height of parallelogram ABCD as well as triangle ABE?

Thanks,
Sri

Attachments

: Parallelogram

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Source: Beat The GMAT — Problem Solving | Sat Nov 23, 2013 11:03 am

riz_gmat: Junior | Next Rank: 30 Posts; Posts: 20; Joined: Thu Sep 26, 2013 9:05 pm; Thanked: 5 times

by riz_gmat » Sat Nov 23, 2013 12:11 pm

If we join diagonal BD of parallelogram ABCD , we get another triangle ABD, such that
Area of || gram ABCD = 2 Area of tri (ABD) ( since both have same base and lie bet same parallel lines) ---- 1
Area of || gram ABCD = 3 Area of tri (ABE) (Given in the question)---- 2

From 1 & 2,
2 Area of tri (ABD) = 3 Area of tri (ABE)
ie, Area of tri (ABE) / Area of tri (ABD) = 2 / 3

tri (ABE) and tri (ABD) are Similar triangles (AA Test of similarity)

Hence , Area of tri (ABE) / Area of tri (ABD) = (AE)^2 / (AD)^2 = 2/3
Therefore, [spoiler]AE / AD = sqaure root of ( 2 / 3)[/spoiler]
whatz the answer?

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Nov 23, 2013 2:53 pm

gmattesttaker2 wrote: In parallelogram ABCD, point E lies on side AD. If the area of the parallelogram
is three times the area of triangle ABE, what is the ratio of the length of AE to
the length of AD?

ABCD can be ANY type of parallelogram.

ABCD:
Let ABCD be a square where s=3.
Area of square ABCD = sÂ² = 3Â² = 9.

âˆ†ABE:
Since the area of square ABCD is 3 times the area of âˆ†ABE, the area of âˆ†ABE = 3.
Thus:
(1/2)(AE)(AB) = 3
(1/2)(AE)(3) = 3
AE = 2.

Resulting ratio:
AE : AD = 2:3.

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