Show that the line segments joining the midpoints of the opposite sides of a quadrilateral and bisect each other.

Given:- ABCD is a quadrilateral

Formula used:- Line joining midpoints of 2 sides of triangle

Is parallel and half of 3^rd side

Solution:-

BD is diagonal of quadrilateral

EH is the line joined by midpoints of triangle ABD,

∴EH is parallel and half of BD

GF is line joined by midpoints of side BC&BD of triangle BCD

∴GF is parallel and half BD

⇒ If HE is parallel to BD and BD is parallel to GF

∴ It gives HE is parallel to GF

⇒ If HE is half of BD and GF is also half of BD

∴ It gives HE is equal to GF

⇒ AC is another diagonal of quadrilateral

GH is the line joined by midpoints of triangle ADC,

∴GH is parallel to AC

∴GH is half of AC

FE is line joined by midpoints of side BC&AB of triangle ABC

∴FE is parallel to AC

∴ FE is half of AC

⇒ If GH is parallel to AC and AC is parallel to FE

∴ It gives GH is parallel to FE

⇒ If GH is half of AC and FE is also half of AC

∴ It gives GH is equal to FE

If both opposite sides are parallel and equal

Then, the quadrilateral is parallelogram

If EFGH is parallelogram

Then their diagonal bisect each other

If diagonal of parallelogram is the line joining midpoint of opposite sides of quadrilateral

Then;

Line joining midpoints of opposite sides of quadrilateral bisect each other.

Conclusion:-

Lines joining midpoints of opposite sides of quadrilateral bisect each other