Question 62348: Please help! Problem: "Prove that the figure formed by connecting the midpoints of the sides of any quadrilateral is a parallelogram. Hint: a line segment connecting the midpoints of 2 sides of a triangle is parallel to the 3rd side."
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Please help! Problem: "Prove that the figure formed by connecting the midpoints of the sides of any quadrilateral is a parallelogram. Hint: a line segment connecting the midpoints of 2 sides of a triangle is parallel to the 3rd side.
LET ABCD BE THE QUADRILATERAL AND E,F,G,H BE THE MIDPOINTS OF AB,BC,CD,DA.
JOIN EFGH.JOIN THE DIAGONAL AC.
IN TRIANGLE ABC,
E IS MID POINT OF AB AND F IS THE MID POINT OF BC.....GIVEN
HENCE EF||AC AND
EF=AC/2......AS PER THEOREM.
SIMILARLY IN TRIANGLE ADC ,
G IS MIDPOINT OF CD AND H IS MIDPOINT OF AD.
HENCE GH||AC AND
GH = AC/2..AS PER THEOREM.
HENCE EF=GH.
IN THE FIGURE EFGH,EF||GH AND EF=GH.
HENCE EFGH IS A PARALLELOGRAM
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