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In parallelogram EFGH, let M be the midpoint of side EF, and let N be the midpoint of side EH. Line segments FH and GM intersect at P, and line segments FH and GN intersect at Q. Find PQ/FH. You don't need to give an answer if you don't have one. A hint or a diagram would be just as helpful. Thanks!

 May 28, 2020
 #1
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In parallelogram EFGH, let M be the midpoint of side EF, and let N be the midpoint of side EH. Line segments FH and GM intersect at P, and line segments FH and GN intersect at Q. Find PQ/FH. You don't need to give an answer if you don't have one. A hint or a diagram would be just as helpful. Thanks!

 

Here is a link to an adjustable graph. I do not remember sharing a link to a GeoGebra graph before so I will be interested in whether it works.  I suppose it will....

 

 

https://www.geogebra.org/classic/wbfqyt8g

 

I haven't worked an answer yet.

 May 28, 2020
edited by Melody  May 28, 2020
 #2
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The link works Melody.

Alan  May 28, 2020
 #3
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Thanks Alan :)

Melody  May 28, 2020
 #4
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I still have not worked out how to do the general case, however, a square is a special case of a parallelogram.

The question implies that the answer will always be the same.

so...

 

If you look at this specific case (A square of side length 2)

distance HF is easy to work out

then work out the coordinates of Q

then distance OQ can be found.

 

Now you can finish it.

 

 

 May 28, 2020
edited by Melody  May 28, 2020
 #5
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Thanks Melody! I think I have the answer now.

 May 28, 2020
 #6
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Did you do it with the square?

Melody  May 29, 2020
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Yes! Good to realize in this situation we can assume it is a square and then know it is true.

Guest May 30, 2020
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We can't really.

We certainly have not proven that this is a general ratio.

However, the question did not ask us to prove anything and it implied that the answer would always be the same.

So it follows that if we find it for a specific case then we will have the answer that is being asked for.

Melody  May 30, 2020

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