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determine the co-ordinates of the midpoint x of side AB and midpoint Y of side AC of ABC. Find XY and BC and verify that XY=1/2 Bc.

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I have a question about your question. I believe you are leaving out some information needed to solve this question. 
 
I will state that xy cannot be // to bc because they are midpoints on the same line. They do have the same slope however, if that is what is needed. 
 
It looks as if you are missing the coordinates of A, B and C. 
 
Thank you for the coordinates, now I can help you!
 
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So let's find the 2 midpoints first.
 
Part 1: The midpoint formula is (x1 + x2/2 , y1 + y2/2).
 
Use the points for A and B to find X
 
X = (4+0/2 , 6+0/2) = (4/2 , 6/2) = (2, 3)
 
Use the points for A and C to find Y
 
Y = (8+0/2, 2+0/2) = (8/2 , 2/2) = (4, 2)
 
Now we need to find out if a line drawn between X and Y is // to a line between B and C.
 
Part 2: The slope formula is: m = (y2 - y1 / x2 - x1)
 
mXY = 3-1/2-4 = 2/-2 = -1
mAB = 2-6/8-4 = -4/4 = -1
 
Because the slopes are the same, XY // BC.
 
Part 3: Now to prove that XY = 1/2 BC
 
For this, you have to use the distance formula between the two points of each line and compare the answers. 
 
XY = √(4-2)2 + (3-1)2 = √22 + 22 = √8 = 2√2
BC = √(8-4)2 + (6-2)2 = √42 + 42 = √32 = 4√2
 
Is XY half of BC? 1/2 (4√2) = 2√2 Yes.
 
I hope this helps.
 

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