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in Quadrilaterals by (58.8k points)

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

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Construct lines KH, BD and GL

We know that K and H are the midpoints of AD and AB

Consider △ ABD

Using the mid-point theorem

We get

KH = ½ BD

Consider △ CBD

Based on the midpoint theorem

GL = ½ BD

So we get

KH = GL

Consider △ KOH and △ GOL

We know that KH = GL

From the figure we know that alternate angles are equal

So we get

∠ OKH = ∠ GLO and ∠ OHK = ∠ OGL

By SAS congruence criterion

△ KOH ≅ △ GOL

OG = OH and OK = OL (c. p. c. t)

So we know that GH and KL bisect each other.

Therefore, it is proved that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

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