Gregmat full test 1 I Doubt#2

GRE Quant Problem Solving
1

Hi! Did not quite get the explanation of this question-

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Here, Greg says that to maximise this quad., we need to make it a square as much as possible- so by that logic, should not we assume sides as either 5X5 or 7.5X7.5?

Did not really understand this solution. I get that square will allow us to maximise, but what is the logic for choosing mid points of both side lengths? How do we know that is this contruction is a square, it will necessarily cut the sides at their mid points?

2

Hey hi

I have not watched this video but mathematically we should be able to prove that taking midpoints would maximize the area of the rectangle BCHF in this scenario. But before going into that approach, I think the observation the problem expects us to make here is:

  • if we take midpoints and check area of BCHF, it is 37.5 and is already larger than the given quantity B. So definitely the BCDF would be more larger.

Coming to your question:

Why would this construction necessarily cut the sides at their midpoints?
Ans: Start with B being midpoint. We already have that BF || CD, which means ABF is right angled at B as well.

  1. So BF/AB = CE/AC = 15/10 => BF = AB x 15/10
  2. We already know AB = 5. Plugging above: BF = AB x 15/10 = 5 x 15/10 = 7.5.
  3. And this means CH must be 7.5 as well, confirming that the construction gives H at midpoint of CE if B is midpoint of AC.

Hope this helps!

3

yup this is super useful, thank you for giving both the expected inference and answering my question!