Question 1124582: Points A(-1,0),B(3,7),and C(5,-2) are the midpoints of the sides YZ, ZX AND XY of triangle XYZ. find the equation of the line XY.
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
Assuming that the given points are midpoints of the RESPECTIVE sides of the triangle, A is the midpoint of YZ, B is the midpoint of XZ, and C is the midpoint of XY.
Any line joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half the length of that third side. That means
(1) YZ is parallel to BC and twice the length of BC;
(2) XZ is parallel to AC and twice the length of AC; and
(3) XY is parallel to AB and twice the length of AB.
From A to B is 4 units right and 7 units up; so point X is 4 units right and 7 units up from C, and point Y is 4 units left and 7 units down from C.
From A to C is 6 units right and 2 units down; so point X is 6 units right and 2 units down from B, and point Z is 6 units left and 2 units up from B.
From B to C is 2 units right and 9 units down; so point Y is 2 units right and 9 units down from A, and point Z is 2 units left and 9 units up from A.
ZA and AY are parallel to and the same length as BC
ZB and BX are parallel to and the same length as AC
YC and CX are parallel to and the same length as AB
It should be easy to determine the coordinates of X, Y, and Z.
Of course, the problem doesn't require us to do all that. It only asks for the equation of line XY; and we can find that with far less work.
I only demonstrated the process so you will be able to solve similar problems where you are asked to find the vertices of triangle XYZ.
Using ideas discussed above, we know the line XY passes through C(5,-2) and has the same slope as AB.
The slope of AB is 7/4 ("right 4, up 7" from above). So one form of the equation of line XY is
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