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GRE Geometry (Select One) http://www.800score.com/forum/viewtopic.php?f=12&t=3205 
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Author:  questioner [ Tue Jul 20, 2010 11:28 am ]  
Post subject:  GRE Geometry (Select One)  
In the figure above, lines L and P are parallel. The segment AD is the same length as the segment DC; AD is parallel to BC. If the length of AD is equal to 4, and angle ADC is equal to 60º, what is the area of ABCD? A. 4√2 B. 4√3 C. 8√2 D. 8√3 E. 16√2 (D) AB is parallel to DC and AD is parallel to BC, so ABCD is parallelogram. We know how to find the area of a parallelogram we multiply the height of parallelogram by its base length. We are told that the angle ADC is a 60º angle. If we draw an altitude from A straight down and perpendicular to line P, the length of that altitude will be equal to the height of the rhombus. Moreover, it forms a 603090 triangle, so we can easily find its length. Since the hypotenuse of that triangle is equal to 4, the second longest side must be equal to 2√3 (since the proportions for the triangle run x, x√3, and 2x). To find the area of a rhombus, multiply the height of the rhombus by its base length: 4 × 2√3 = 8√3, or answer choice (A).  The sum of the interior angles should be (n – 2) × 180. So the sum of the interior angles for ABCD = 360⁰. How is it determined that the traingle formed from A is a 306090 triangle?

Author:  Gennadiy [ Tue Jul 20, 2010 11:39 am ]  
Post subject:  Re: math (test 1, question 9): geometry  
We determine that triangle EDA is a 906030 triangle because: angle ADC is 60 degrees (we know that from the question statement) angle AED is 90 degrees because AE is height of parallelogram ABCD angle EAD is 30 degrees since it equals 180⁰ – angle AED  angle ADC = 180⁰ – 90⁰ – 60⁰ = 30⁰

Author:  questioner [ Thu Aug 12, 2010 8:06 am ] 
Post subject:  Re: math (test 1, question 9): geometry 
How did you deduce that the parallelogram formed its special case (i.e., a rhombus)? Given the information from the question and the fact that the figure is not drawn to scale, I find it hard to deduce that the figure is a rhombus. 
Author:  Gennadiy [ Thu Aug 12, 2010 8:50 am ]  
Post subject:  Re: math (test 1, question 9): geometry  
Question statement tells us: "In the figure above, lines L and P are parallel. ... AD is parallel to BC." So we know that ABCD is a parallelogram. As any parallelogram its opposite sides are equal. So AB = DC and AD = BC. Besides, "the segment AD is the same length as the segment DC". So now all 4 sides are equal. Therefore ABCD is rhombus.

Author:  questioner [ Fri Feb 24, 2012 7:00 pm ] 
Post subject:  Re: math (test 1, question 9): geometry 
I think the answer should be 16√3. As area of each triangle of the parallelogram would be 1/2 × 4 × 4 × sin(60⁰) = 8√3. 
Author:  Gennadiy [ Fri Feb 24, 2012 7:03 pm ] 
Post subject:  Re: math (test 1, question 9): geometry 
questioner wrote: I think the answer should be 16√3. As area of each triangle of the parallelogram would be 1/2 × 4 × 4 × sin(60⁰) = 8√3. sin(60⁰) = √3/2.

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