Question

If the mid-points of the consecutive sides of a quadrilateral are joined, then show (by using vectors) that they form a parallelogram.​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Given: If the mid-points of the consecutive sides of a quadrilateral are joined.

To show: (by using vectors) that the midpoints form a parallelogram.​

Solution:

Assume that, ABCD is the quadrilateral and M,N,O,P are the mid points of the sides AB, BC, CD, DA respectively.

Know that, position vectors of M,N,O,P are $\[\frac{{\overrightarrow a + \overrightarrow b }}{2},\frac{{\overrightarrow b + \overrightarrow c }}{2},\frac{{\overrightarrow c + \overrightarrow d }}{2},\frac{{\overrightarrow d + \overrightarrow a }}{2}\]$respectively.

Understand that to show that MNOP is a parallelogram, $\[\overrightarrow {MN} = \overrightarrow {PO} ,\overrightarrow {MP} = \overrightarrow {NO} \]$should be proved first.

Therefore, Prove that, $\[\overrightarrow {MN} = \overrightarrow {PO} \]$

$\[\begin{array}{l}\overrightarrow {MN} = \overrightarrow {PO} \\ \Rightarrow \frac{{\overrightarrow b + \overrightarrow c }}{2} - \frac{{\overrightarrow a + \overrightarrow b }}{2} = \frac{{\overrightarrow c + \overrightarrow d }}{2} - \frac{{\overrightarrow d + \overrightarrow a }}{2}\end{array}\]$

$\[ \Rightarrow \frac{{\overrightarrow b + \overrightarrow c - \overrightarrow a - \overrightarrow b }}{2} = \frac{{\overrightarrow c + \overrightarrow d - \overrightarrow d - \overrightarrow a }}{2}\]$

$\[\begin{array}{l} \Rightarrow \frac{{\overrightarrow c - \overrightarrow a }}{2} = \frac{{\overrightarrow c - \overrightarrow a }}{2}\\ \Rightarrow LHS = RHS\end{array}\]$

$\[ \Rightarrow \overrightarrow {MN} ||\overrightarrow {PO} \]$

Prove that, $\[\overrightarrow {MP} = \overrightarrow {NO} \]$.

$\[\begin{array}{l}\overrightarrow {MP} = \overrightarrow {NO} \\ \Rightarrow \frac{{\overrightarrow d + \overrightarrow a }}{2} - \frac{{\overrightarrow a + \overrightarrow b }}{2} = \frac{{\overrightarrow c + \overrightarrow d }}{2} - \frac{{\overrightarrow b + \overrightarrow c }}{2}\end{array}\]$

$\[ \Rightarrow \frac{{\overrightarrow d + \overrightarrow a - \overrightarrow a - \overrightarrow b }}{2} = \frac{{\overrightarrow c + \overrightarrow d - \overrightarrow b - \overrightarrow c }}{2}\]$

$\[\begin{array}{l} \Rightarrow \frac{{\overrightarrow d - \overrightarrow b }}{2} = \frac{{\overrightarrow d - \overrightarrow b }}{2}\\ \Rightarrow LHS = RHS\end{array}\]$

$\[ \Rightarrow \overrightarrow {MP} ||\overrightarrow {NO} \]$

Observe that, $\[ \overrightarrow {MP} ||\overrightarrow {NO} \]$and $\[ \overrightarrow {MN} ||\overrightarrow {PO} \]$.

Therefore, the MNOP is a parallelogram.

Hence proved.