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Parallelogram

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by jfranco23 » Mon Mar 16, 2009 4:37 pm
Each side of a certain parallelogram has lenght 6. If the area of the parallelogram is 18, which of the following is the measure of one of its angles?

A. 30
B. 45
C. 60
D. 90
E. 120
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by m&m » Mon Mar 16, 2009 5:05 pm
if the parallelogram was a square the area would have been 36 (and the area would have been 90. So 90 is out.

we can create 2 equal triangles and 1 rectangle in the parallelogram. where the base of the triangle is x say.

so we know area of rectangle = (6-x)(6^2-x^2)^0.5
and area of each triangle is = 0.5*(x)(6^2-x^2)^0.5

so total area = area rect+2*area of triangle
(6^2-x^2)^0.5 [6-x+x]=18
(6^2-x^2)^0.5 = 3
(6^2-x^2) = 9
27 = x^2
x=3*sqrt(3)

so angles of triangle form a 30-60-90 relationship with 60 opposite to x. So angles are either 30 or 90+60=150 hence A
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by crackthetest » Mon Mar 16, 2009 6:45 pm
The height is 3, since area = b*h ( 18=6h )

drop perpendiculars to get two triangles.
the hypotenuse is 6 and height (of parallelogram) is 3. this forms a 30-60-90 triangle.

answer is a. ( since the parallelogram forms a 150-30-150-30 angle).
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