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November 13th, 2007, 12:34 PM  #1 
Newbie Joined: Nov 2007 Posts: 3 Thanks: 0  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.

November 14th, 2007, 10:32 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,154 Thanks: 1421 
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? 
June 13th, 2008, 07:01 PM  #3 
Newbie Joined: Jun 2008 From: New Zealand Posts: 20 Thanks: 0  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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