Lines joining midpoints of quadrilaterals?????

Question

prove that the lines joining the midpoints of opposite sides of a quadrilateral bisect each other. (a line or segment bisects another segment if it divides the segment into 2 equal parts) Hint: Let the vertices of the quadrilateral be (0,0), (a,b), (c,d), (e,0)

i need solutons... thanks!

Answer 1

The vertices are A(0,0), B(a,b), C(c,d) and D(e,0)

Midpoint of AB is (a/2, b/2) and midpoint of CD is [(c+e)/2, d/2]

The midpoint of these two points is [(a+c+e)/4, (b+d)/4]

Midpoint of BC is [(a+c)/2, (b+d)/2] and of AD is (e/2, 0)

The midpoint of these two points is [(a+c+e)/4, (b+d)/4]

Thus, the lines joining the midpoints of opposite sides of a quadrilateral  intersect at their midpoints and hence bisect each other.

Thubs up to bskelkar for answering correctly.

Answer 2

Simple! Let the vertices of the quadrilateral be A(0,0), B(a,b), C(c,d),

D(e,0)

So midpoint P of AB is (a/2,/b/2), Q of BC is ((a+c)/2, (b+d)/2)), R of CD is ((c+e)/2, d/2) and S of DA is (e/2, 0)

Now midpoint X of PR is ({a+c+e}/4, {b+d}/4)

                    Y of QS is same as X so PR and QS bisect each other.