Question

Show that the line segments joining the mid-point of the opposite sides of a

quadrilateral bisect each other.​

Answer 1

Basic Concept Used ;-

1. Midpoint Theorem :-

This theorem states that a line segment joining the midpoint of two sides of a triangle is parallel to third side and equals to half of it.

2. If in a quadrilateral, one pair of opposite sides are equal and parallel, then quadrilateral is a parallelogram.

3. In parallelogram, diagonals bisect each other.

Solution :-

Let us assume a quadrilateral ABCD such that P, Q, R, S are the midpoints of the side AB, BC, CD, DA respectively.

☆ Construction :- Join PQ, QR, RS, SP and AC

☆ Now, In triangle ABC,

• P is the midpoint of AB

• Q is the midpoint of BC.

☆ So, By midpoint theorem,

• PQ || BC and PQ = 1/2 AC -------(1)

☆ Now, In triangle ACD

• R is the midpoint of CD

• S is the midpoint of AD

☆ So, by midpoint theorem,

• RS || AC and RS = 1/2 AC ---------(2)

☆ From equation (1) and equation (2), we concluded that

• ⇛ PQ || RS and PQ = RS

• ⇛ PQRS is a parallelogram.

• ⇛ Diagonal PR and Diagonal QS bisects each other.

Hence,

• Line segment joining the mid-point of the opposite sides of a quadrilateral bisect each other.

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

1. The quadrilateral formed by joining the mid points of a rectangle is a rhombus.

2. The quadrilateral formed by joining the mid points of a rhombus is a rectangle.

3. The quadrilateral formed by joining the mid points of a square is a square.