Premium Online Home Tutors
3 Tutor System
Starting just at 265/hour

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


Answer :

Let ABCD is a quadrilateral and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.

\(\Rightarrow \) AS = SD, AP = BP, BQ = CQ and CR = DR.

We have to show that: PR and SQ bisect each other i.e., SO = OQ and PO = OR.
image

Now, in \(\triangle{ADC}\), S and R are mid-point of AD and CD.

We know that, the line segment joining the mid-points of two sides of a triangle is parallel to the third side.

Thus, by midpoint theorem,
SR || AC and SR = (\(\frac{1}{2} \)) AC ...(i)

Similarly, in \(\triangle{ABC}\), P and Q are mid-point of AB and BC.


Thus, by midpoint theorem,
PQ || AC and PQ = (\(\frac{1}{2} \) ) AC ...(ii)
From Eq. (i) and (ii), we get,
PQ || SR and PQ = SR = (\(\frac{1}{2} \) ) AC

Therefore, Quadrilateral PQRS is a parallelogram whose diagonals SQ are PR.

Also, we know that diagonals of a parallelogram bisect each other. So, and bisect each other.
Hence, it is proved that PR and SQ bisect each other.

NCERT solutions of related questions for Quadrilaterals

NCERT solutions of related chapters class 9 maths

NCERT solutions of related chapters class 9 science