Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

Solution:

Given, ABCD is a trapezium

AC and BD are the diagonals of the trapezium

M and N are the midpoints of the diagonals AC and BD

We have to prove that MN || AB || CD

Join CN and extend it to meet AB at E.

Considering triangles CDN and EBN,

Since N is the midpoint of BD

DN = BN

We know that the alternate interior angles are equal.

∠DCN = ∠NEB

∠CDN = ∠NBE

The ASA congruence rule states that if any two consecutive angles of a triangle along with a non-included side are equal to the corresponding consecutive angles and the non-included side of another triangle, the two triangles are said to be congruent.

By ASA criterion, the triangles CDN and EBN are congruent.

By CPCTC,

DC = EB

CN = NE

Considering triangle CAE,

M and N are the midpoints of AC and CE

MN || AE

By midpoint theorem,

MN || AB || CD

Therefore, it is proved that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

✦ Try This: If PQRS is trapezium such that PQ > RS and L, M are the mid-points of the diagonals PR and QS respectively then what is LM equal to?

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8

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NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 17

Prove that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

Summary:

It is proven that the line joining the mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium

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