SOLUTION: Verify the quadrilateral formed by joining midpoint of sides of a quadrilateral is a parallelogram's
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Question 928077: Verify the quadrilateral formed by joining midpoint of sides of a quadrilateral is a parallelogram's
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I assume that what is expected is an analytical geometry proof.
Let the quadrilateral be OABC, and let the vertices and their coordinates be
,
,
, and
.
Those assumptions pose no restrictions, because
I can call the vertices any letter I chose;
I can always place my system of coordinates with one vertex is at the origin, and another vertex on the x-axis, and
all coordinates could be divided by 2 to make my , , , , and .
The midpoint of OA is ---> .
The midpoint of AB is ---> .
The midpoint of BC is ---> .
The midpoint of OC is ---> .
PQRS is a parallelogram if PQ is parallel to RS and PS is parallel to QR.
The slope of PQ is ,
and the slope of RS is ,
so PQ and RS are parallel.
The slope of PS is ,
and the slope of QR is ,
so PS and QR are parallel.
Since PQ is parallel to RS and PS is parallel to QR, PQRS is a parallelogram.
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