Is it necessary to construct the pairs of arcs above and below XY? Instead, can we construct both the pairs of arcs on the same side of XY? Explore this through construction, and then justify your answer.
Yes, both pairs of arcs can be constructed on the same side of XY.
Justification: The perpendicular bisector is determined by any two points that are equidistant from both endpoints X and Y. When arcs of equal radius are drawn from X and Y, their intersection points (whether both above XY or both below XY) are each equidistant from X and Y. ∴ The line joining any two such intersection points will be the perpendicular bisector of XY.
Explanation
The chapter establishes that any point equidistant from X and Y lies on the perpendicular bisector. When we draw arcs from X and Y with the same radius, each intersection point is equidistant from both X and Y. The perpendicular bisector is a unique line, and any two points on it are sufficient to draw the line. Therefore, whether the two intersection points are both above XY, both below XY, or one above and one below XY, joining them will always give the perpendicular bisector.
Solution Steps
The perpendicular bisector is determined by any two points that are equidistant from both endpoints X and Y.
When arcs of equal radius are drawn from X and Y, their intersection points are each equidistant from X and Y.
The line joining any two such intersection points will be the perpendicular bisector of XY.