State whether the following statements are true or false. Justify your answer.
The point $ \mathrm{A}(2,7) $ lies on the perpendicular bisector of line segment joining the points $ P(6,5) $ and $ Q(0,-4) $.

Given :

The given statement is,

The point \( \mathrm{A}(2,7) \) lies on the perpendicular bisector of 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 the point $A$ 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,

Here,

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

Therefore,

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

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

$PA = \sqrt{16+4}$

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

$PA = 2\sqrt{5}$.

The distance between the points A and Q is,

Here,

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

Therefore,

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

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

$AQ = \sqrt{4+121}$

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

$AQ = 5\sqrt{5}$.

PA is not equal to AQ.

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

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Updated on: 10-Oct-2022

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