The point \(A(2,7)\) lies on the perpendicular bisector of the segment joining \(P(6,5)\) and \(Q(0,-4)\).
False.
To check if a point lies on the perpendicular bisector of a line segment, we must test whether the point is equidistant (same distance) from both end points of the segment.
Step 1: Write the coordinates of all points.
Step 2: Use the distance formula:
\( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Step 3: Calculate distance \(AP\).
\[ AP = \sqrt{(2 - 6)^2 + (7 - 5)^2} = \sqrt{(-4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} \]
Step 4: Calculate distance \(AQ\).
\[ AQ = \sqrt{(2 - 0)^2 + (7 - (-4))^2} = \sqrt{(2)^2 + (11)^2} = \sqrt{4 + 121} = \sqrt{125} \]
Step 5: Compare the two distances.
We have \(AP = \sqrt{20}\) and \(AQ = \sqrt{125}\).
Conclusion: Since \(AP \neq AQ\), point \(A\) is not equidistant from \(P\) and \(Q\). Therefore, \(A\) does not lie on the perpendicular bisector of the line segment \(PQ\).