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Geometry & Algebra combined (Parallelogram)

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Geometry & Algebra combined (Parallelogram)

by euro » Fri Sep 17, 2010 11:14 am
(Q) The length of one of the sides of a parallelogram is 'x inches'. The perimeter and area of the parallelogram are 'y inches' and 'z sq. inches' respectively.
Is 2x^2 - xy + 2z = 0 ?

(1) One of the angles of the parallelogram is a right angle.
(2) One side of the parallelogram is half of the adjacent side.


Official Answer is A
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Source: — Data Sufficiency |
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by aimscore » Fri Sep 17, 2010 1:18 pm
Hi,

Lets assume that the sides of the parallelogram are x inches, e inches resp.
Lets assume that height of the parallelogram is h inches (although this info is redundant as u will see further)
Hence the perimeter of the parallelogram y= 2(x+e) inches.
Area of the parallelogram z = e*h sq inches. (Assuming that e is the base)

Now with statement 1, we get that "One of the angles of the parallelogram is a right angle".
This means that your parallelogram is now a rectange !!
The perimeter still remains y=2(x+e) inches.
Area z will now become , z= xe sq inches.

So we now have 2 equations,
ie y=2(x+e), y= 2x+2e
z= xe

Back to the question:

2x^2 - xy +2z=0
Substitute the values we have in the above equn,

2x^2 - x(2x+2e) + 2(xe)
2x^2 - 2x^2 - 2xe +2ex

Each of the terms cancel out leaving you 0:):) Hence Data is sufficient
Hence your answers now narrow down to A,D


With statement 2, "One side of the parallelogram is half of the adjacent side", the quadrilateral still remains a parallelogram and we would need the "height h" to solve the equation.
Hence the data for statement 2 is insufficient. Hence D is eliminated.

Right answer is therefore A:)
Hope this helps.
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