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Question 1197814: Hi can you help me asnwer this question, with a solution and its formula.
Given a quadrilateral ( -7, 2 ) ( -5, -8 ), ( 3, -9 ) and ( 4, -3 ) show that the line segments joining the midpoints of the opposite sides bisect each other.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
We are given these four vertices of the quadrilateral.
A = (-7, 2)
B = (-5, -8)
C = (3, -9)
D = (4, -3)
Define the following midpoints
P = midpoint of side AB
Q = midpoint of side BC
R = midpoint of side CD
S = midpoint of side DA
Drawing:
I'll show the steps of how to locate point P using the midpoint formula.
The steps for midpoints Q, R, and S will be left for the student to do.
The x coordinates of point A(-7,2) and B(-5,-8) are -7 and -5 respectively.
Add them up: -7+(-5) = -12
Cut the result in half: -12/2 = -6
This is the x coordinate of point P.
Repeat for the y coordinates.
Add: 2+(-8) = -6
Cut in half: -6/2 = -3
This is the y coordinate of point P.
Here's where all four midpoints are located
P = (-6, -3)
Q = (-1, -8.5)
R = (3.5, -6)
S = (-1.5, -0.5)
The decimal values are exact.
Based on how the midpoints are labeled, we have these pairs of opposite midpoints:Those form segments PR and QS respectively.
The equation of line PR is 6x + 19y = -93
The equation of line QS is 32x + 2y = -49
The scratch work for determining the equation of line PR is shown here.
Follow similar steps to determine the equation of line QS.
We have this system of equations
Use elimination, substitution, graphing technology, or Cramer's Rule to solve that system of two equations.
I'll skip the steps for this subsection as this post is getting fairly lengthy as it is.
Let me know if you get stuck here and I'll update the page.
The solution to that system of equations is (x,y) = (-1.25, -4.5) which I'll label as point E.
The task now is to prove these two claimsUse the distance formula to do so.
I'll show the steps to computing the length of segment PE.
(x1,y1) = P = (-6, -3)
(x2,y2) = E = (-1.25, -4.5)
This distance is exact.
Therefore, segment PE is exactly  units long.
Your goal is to show that segment ER is also exactly units long through the distance formula.
Once you've shown that PE = ER, it directly confirms that segment PR has been bisected (i.e. cut in half) at point E.
Segments SE and EQ will be equal to their own different value. I'll let the student compute these segment lengths.
After you've shown that SE = EQ, then that directly leads to confirmation that segment QS has been bisected at point E.
Answer by ikleyn(52887)  (Show Source):
You can put this solution on YOUR website! .
There are two facts (theorems) of Geometry that are valid for any convex quadrilaterals.
(1) For any convex quadrilateral, the segments joining its midpoints form a parallelogram.
(2) For any parallelogram, its diagonals bisect each other.
Regarding n.1, I proved it for you just twice in my preceding posts - you may find the proofs there.
Regarding n.2, everybody knows this standard property, who is familiar with the term of " parallelogram ".
So, the statement of your post is a direct and straightforward consequence of general properties n.1 and n.2.
When you try to associate it with concrete coordinates of vertices, you only embarrass a reader,
forcing him (or her) to perform unnecessary computational job with coordinates of the vertices.
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