Geometry

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Geometry

by knight247 » Tue Dec 13, 2011 6:04 am
OA is D

I solved it in this manner... Point F with co-ordinates 2,-1 is the midpoint of two points... i.e.
x1+x2/2=2 or x1+x2=4 and y1+y2/2=-1 or y1+y2=-2. I then checked which answer choices meet this requirement. I narrowed it down to B and D. But am unable to go any further. Hoping to get some detailed explanations. Cheers.
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by Elena89 » Tue Dec 13, 2011 6:30 am
To save time, one can tell from just looking at the answer choices.

Its obvious from the figure that if y-coordinate of F lies in the negative y-axis then B being further below would also lie in the -ve y-axis, so this cancels out the first 3 answer choices.
Now again looking at the coordinates of A in the answer choices (D) and (E), we know that if F has negative y-coordinate of -1 then that of point A must be greater than -1 (i.e in the positive y-axis)
And answer choice (D) solves that issue. so that's the correct one..

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by GMATGuruNY » Tue Dec 13, 2011 6:53 am
knight247 wrote:OA is D

I solved it in this manner... Point F with co-ordinates 2,-1 is the midpoint of two points... i.e.
x1+x2/2=2 or x1+x2=4 and y1+y2/2=-1 or y1+y2=-2. I then checked which answer choices meet this requirement. I narrowed it down to B and D. But am unable to go any further. Hoping to get some detailed explanations. Cheers.
ALWAYS LOOK AT THE ANSWER CHOICES.

Since B and D include the same two coordinate pairs, just in reverse order, the ordering in the answer choices must reflect the ordering in the question: the first coordinate pair in each answer choice is B, the second A.
Since the x coordinate of F is 2, the x coordinate of B must be greater than 2, and the x coordinate of A must be less than 2.
Only answer choice D works: (6,-3) and (-2,1).

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by GmatMathPro » Tue Dec 13, 2011 7:01 am
When the midpoints of the sides of triangles are connected like this, two important things happen. 1) The segment connecting the midpoints is parallel to the third side. 2) The segment connecting the midpoints is half the length of the third side. These properties are a result of similar triangles that can be proved by the ~SAS similarity postulate.

Note that going from point D(1,3) to point E(-3,5) you go up 2 units and 4 to the left. From the abovve we can deduce that FA=ED and that FA is parallel to ED. We can preserve these two properties only if we also go up 2 and left 4 to get to A. F is at (2,-1), so going up 2 and left 4 puts you at (-2,1) for point A. Through similar logic we can get to point B by going right 4 and down 2 from (2,-1) to get to (6,-3) for point B.