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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