Question

Show that the lines joining the mid-points of opposite sides of a quadrilateral bisect each other.

Answer 1

Let P, Q R and S are the mid-points of the sides AB, BC, CD and DA respectively of `squareABCD.` Join AC.,
Since, P and Q are the mid-points of BA and BC respectively.
`thereforePQ"||"ACand PQ=1/2AC" "("mid-point theorem")" "...(1)`
In `DeltaDAC,`
Since, R and S are th mid-points of DC and DA respectively.
`thereforeRS"||"ACand RS=1/2AC" "("mid-point theorem")" "...(2)`
From equations (1) and (2)
`PQ"||"RSand PQ=RS`
`impliessquarePQRS` is a parallelogram `" "("thereforeone pair of opposite side is equal and parallel")`
We know that the diagonals of a parallelogram bisect each other.
`therefore` PR and QS bisect each other.