Mathematics
Prove that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Mid-point Theorem
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Answer
Let ABCD be the quadrilateral and E, F, G and H be the mid-point of AD, AB, BC and CD.
By mid-point theorem,
The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it.
Let ABCD be the quadrilateral and E, F, G and H are the mid-points of sides AD, AB, BC and CD.
In △BCD,
Since, G and H are the mid-points of BC and CD respectively.
⇒ GH || BD and GH = BD …(1)
In △BAD,
Since, F and E are the mid-points of AB and AD respectively.
⇒ FE || BD and FE = BD …(2)
From eq.(1) and (2), we have :
⇒ GH = FE and FE || GH
∴ EFGH is a parallelogram.
We know that,
Diagonals of the parallelogram, bisect each other.
EG and FH are the diagonals of parallelogram EFGH bisects each other.
Hence, proved that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.
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