In the adjoining figure, ABCD is a trapezium in which AB || DC. If M and N are the mid-points of AC and BD respectively. Prove that MN = (AB - CD).

Question

KnowledgeBoat · Accepted Answer

From figure, AB // DC EB // DC In △BNE and △CND, ⇒ ∠BNE = ∠CND (Vertically opposite angles are equal) ⇒ ∠BEN = ∠NCD (Alternate angles are equal) ⇒ BN = DN (N is the mid-point of BD) ∴ △BNE ≅ △CND ⇒ BE = CD (Corresponding parts of congruent triangles are equal) ⇒ NE = CN (Corresponding parts of congruent triangles are equal) ∴ N is the mid-point of CE. By mid-point theorem, The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it. Since, M and N are the mid-points of AC and CE respectively. MN || AE ⇒ MN = AE ⇒ MN = (AB - BE) ⇒ MN = (AB - CD) (∵ BE = CD) Hence, proved that MN = (AB - CD).

Mathematics

In the adjoining figure, ABCD is a trapezium in which AB || DC. If M and N are the mid-points of AC and BD respectively. Prove that MN = (AB - CD).

Mid-point Theorem

Related Questions

Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rhombus is a rectangle.

Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a square is a square.

In the adjoining figure, ABCD is a trapezium in which AB || DC and E is the mid-point of AD. If EF || AB meets BC at F, show that F is the mid-point of BC.

Two points A and B lie on the same side of a line XY. If AD ⊥ XY and BE ⊥ XY meet XY in D and E respectively and C is the mid-point of AB, show that CD = CE.