State whether the following statements are true or false. Justify your answer.
Points $ \mathrm{A}(4,3), \mathrm{B}(6,4), \mathrm{C}(5,-6) $ and $ \mathrm{D}(-3,5) $ are the vertices of a parallelogram.

Given:

Points \( \mathrm{A}(4,3), \mathrm{B}(6,4), \mathrm{C}(5,-6) \) and \( \mathrm{D}(-3,5) \) are the vertices of a parallelogram.

To do:

We have to find whether the given statement is true or false.

Solution:

The distance between $A(4,3)$ and $B(6,4)$ is,

$A B=\sqrt{(6-4)^{2}+(4-3)^{2}}$

$=\sqrt{2^{2}+1^{2}}$

$=\sqrt{5}$

The distance between $B(6,4)$ and $C(5,-6)$ is,
$B C=\sqrt{(5-6)^{2}+(-6-4)^{2}}$

$=\sqrt{(-1)^{2}+(-10)^{2}}$

$=\sqrt{1+100}$

$=\sqrt{101}$

The distance between $C(5,-6)$ and $D(-3,5)$ is,

$C D =\sqrt{(-3-5)^{2}+(5+6)^{2}}$

$=\sqrt{(-8)^{2}+11^{2}}$

$=\sqrt{64+121}$

$=\sqrt{185}$

The distance between $D(-3,5)$ and $A(4,3)$ is,

$D A=\sqrt{(4+3)^{2}+(3-5)^{2}}$

$=\sqrt{7^{2}+(-2)^{2}}$

$=\sqrt{49+4}$

$=\sqrt{53}$

Here, the lengths of AB, BC, CD and DA are all not equal to each other.

We know that, in a parallelogram opposite sides are equal.

Hence, the given vertices are not the vertices of a parallelogram.