Name: Theorems on a Parallelogram
Uploaded: 2019-07-19
Duration: 12 min 6 s
Description: Learn the theorems on quadrilaterals, such as properties related to the diagonal of a parallelogram and the opposite sides and angles of a parallelogram etc.

Question

Prove that the line segment joining the mid-point of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half the difference of these sides.

Accepted Answer

Let $A B C D$ be the trapezium and $M & N$ are midpoints of $A C & B D$ .

Join $C N$ and produce it to meet $A B$ at $E$ .

$In △ CDN & △ EBN, DN = BN (given) ∠ CND = ∠ ENB (vertically opposite angles) ∠ DCN = ∠ BEN (alternate interior angles) So, △ CDN ≅ △ EBN (AAS congruence) By cpct, CN = EN . . . (1) & DC = EB . . . (2) From (1), N is midpoint of CE \{\}$

So, from midpoint theorem,

$M N ∥ A E \Rightarrow M N ∥ A B &$ we know $A B ∥ C D$ (opposite sides in a trapezium are parallel)

So, $M N ∥ A B ∥ C D$

And $M N = \frac{1}{2} A E$

$M N = \frac{1}{2} (A B - B E)$

From equation $(2)$ ,

$M N = \frac{1}{2} (A B - C D)$

Hence, proved.

Prove that the line segment joining the mid-point of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half the difference of these sides.

Important Questions on Quadrilaterals

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