If a number of circles pass through the end points \(P\) and \(Q\) of a line segment \(PQ\), then their centres lie on the perpendicular bisector of \(PQ\). State True/False and justify.
True.
Step 1: Think of a circle. The centre of a circle is always the same distance from every point on the circle.
Step 2: Here, every circle passes through points \(P\) and \(Q\). So, the centre of the circle must be equal distance (equidistant) from \(P\) and from \(Q\).
Step 3: The set of all points that are equal distance from \(P\) and \(Q\) forms a special line. That line is called the perpendicular bisector of \(PQ\).
Step 4: Therefore, the centre of each circle through \(P\) and \(Q\) must lie on this perpendicular bisector.
Step 5: So, the statement is True.