SOLUTION: I need to prove that a quadrilateral with vertices G(1,1), H(5,3), I(4,5), and J(0,3) is a parallelogram and then that it is a rectangle. For the first part I need to use the midpo

Question 1116609: I need to prove that a quadrilateral with vertices G(1,1), H(5,3), I(4,5), and J(0,3) is a parallelogram and then that it is a rectangle. For the first part I need to use the midpoint formula. For the second part I need to use the rectangle method.
For part 1 I had
(X,y) midpoint= (x1 +x2/2, y1+y2/2)
(X,y)= (1+5/2, y1+3/2)
(X,y)= (3,2) GH
(X,y)= (4+0/2, 5+3/2)
(X,y)=(2,4) IJ
(X,y)= (3+2/2,2+4/2)
(X,y)=(5/2,6/2)
(X,y)=(2.5,3)
For part 2 I had
D=_(x2-x1)^2+(y2-y1)^2
D=_(4)^2+(2)^2
D=16+4
D=20=4 4/9
D=_ (0-4)^2(3-5)^2
D=_ (-4)^2+(-2)^2
D=_16+4
D=_29=4 4/9
The parallelogram method...
The midpoint of GH is (3,2) and the midpoint of IJ is (2,4) so the answer is (#.5,3)
The rectangle method...
The answer is the square root of 20 or 4 4/9
The feedback from my teacher was that I needed the midpoint of the correct sides and a soliD conclusion statement and that I needed two slopes....
I think after typing this out I might have an idea about how to go about this again but I am so confused! and have been trying for days!
Thank you so much for reading this and even if it doesn�t make sense I really appreciate that you offer help!
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
To prove the quadrilateral is a parallelogram, you can use the property that the diagonals bisect each other.
Thus you want to show the midpoints of the two diagonals are the same:
Find the midpoint of JH and the midpoint of GI and compare their coordinates.

To show the parallelogram is a rectangle, it is sufficient to show one angle is a right angle (for a parallelogram with one right angle must have all right angles, hence its a rectangle). To do this, one way is to show that one diagonal and two sides form a right triangle. If, say, |JH|^2 = |GH|^2 + |JG|^2 then you have shown JGH to be a right triangle and you are done.

Recall, the square of the distance between (x1,y1) and (x2,y2) is

Hope this helps...
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As an alternate approach, if you know about slopes (slope = rise / run = (change in y) / (change in x) ), you can compute the slope of GH, HI, IJ, and JG. When the slopes of two lines are equal the lines are parallel. When slopes are negative reciprocals, two lines are perpendicular. This should allow you to quickly prove parallelogram (slope of GH = slope of IJ, slope of JG = slope of HI) and rectangle (slope of GH = -1/(slope of HI)).

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