Whether the following statement is true or false. Justify your answer.

Point A(2,7) lies on the perpendicular bisector of the line segment joining the points P(6,5) and Q$(0, -4)$.

Given :

The given statement is 'Point A(2,7) lies on the perpendicular bisector of the line segment joining the points P(6,5) and Q$(0, -4)$'.

To do :

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

Solution :

If A is lies on the perpendicular bisector of the line PQ, then PA $=$ AQ.

The distance formula is given by, 

$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The distance between the points P and A is,

$(x_1, y_1) = (6, 5)$      $(x_2, y_2) = (2, 7)$

$PA = \sqrt{(2 - 6)^2 + (7 - 5)^2}$

$PA = \sqrt{(-4)^2 + 2^2}$

$PA = \sqrt{16+4}$

$PA = \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

$PA = 2\sqrt{5}$.

The distance between the points A and Q is,

$(x_1, y_1) = (2, 7)$      $(x_2, y_2) = (0, -4)$

$QA = \sqrt{(0 - 2)^2 + (-4 - 7)^2}$

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

$AQ = \sqrt{4+121}$

$AQ = \sqrt{125} = \sqrt{5 \times 25} = 5\sqrt{5}$

$AQ = 5\sqrt{5}$.

PA is not equal to AQ.

Therefore, the given statement is false, because the distance PA and the distance AQ are not equal.