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November 13th, 2007, 11:34 AM   #1
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help please ASAP

Assume that a quadrilateral has the following vertex points: A (a,b) B (c,d) C (e,f) and D (g,h). EFGH are the midpoints of the sides AB,BC,CD,DA. Find the coordinated of E,F,G,H and prove that EFGH is a parallelogram. do not assign munmbers to the points, use the letters.
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November 14th, 2007, 09:32 AM   #2
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E is ((a + c)/2, (b + d)/2). Similar results hold for F, G and H.

What do you know about EF and AC, and similarly about HG and AC?
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June 13th, 2008, 06:01 PM   #3
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Re: help please ASAP

The coordinates of EFGH are as follows
E {(a+c)/2, (b+d)/2}
F {(c+e)/2, (d+f)/2}
G {(e+g)/2, (f+h)/2}
H {(a+g)/2, (b+h)/2}

The slope of line EF Kef is
= [(b+d)/2 - (d+f)/2]/[(a+c)/2 - (c+e)/2]
= (b - f)/(a - e)

The slope of line GH Kgh is
= [(b+h)/2 - (f+h)/2]/[(a+g)/2 - (e+g)/2]
= (b - f)/(a - e)

The slopes of the line EF and GH are the same, so EF and GH are parallel.

For the same reason, EH and FG are parallel.

So EFGH is a parallelogram
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