Question

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that, ABCD is a quadrilateral such that

P is the midpoint of AB

Q is the midpoint of BC

R is the midpoint of CD

S is the midpoint of DA

Now, we have to show that PR and QS bisects each other.

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

Now, In triangle ABC

P is the midpoint of AB

Q is the midpoint of BC

We know

Midpoint Theorem :- This theorem states that line joining the midpoints of the two sides of a triangle is parallel to third side and equals to half of it.

So, By using Midpoint Theorem,

$\rm\implies \:PQ \: \parallel \: AC \: \: and \: \: PQ \: = \: \dfrac{1}{2}AC - - - (1)$

Now, In triangle ACD

R is the midpoint of CD

S is the midpoint of DA

So, By Midpoint Theorem,

$\rm\implies \:RS \: \parallel \: AC \: \: and \: \: RS \: = \: \dfrac{1}{2}AC - - - (2)$

So, from equation (1) and (2), we concluded that

$\rm \: PQ \: = \: RS \: \: and \: \: PQ \: \parallel \: RS$

We know, In a quadrilateral, if opposite pair of sides are parallel and equal, then quadrilateral is a parallelogram.

$\rm\implies \:PQRS \: is \: a \: parallelogram.$

We know,

In parallelogram, diagonals bisects each other.

Hence, PQ and RS bisects each other.

HENCE, PROVED

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MORE TO KNOW

1. The quadrilateral formed by joining the midpoints of the sides of a quadrilateral is a parallelogram.

2. The quadrilateral formed by joining the midpoints of the sides of a square is a square.

3. The quadrilateral formed by joining the midpoints of the sides of a rectangle is a rhombus.

4. The quadrilateral formed by joining the midpoints of the sides of a rhombus is a rectangle.