While constructing one pair of intersecting arcs, is it necessary that we use the same radii for both of them? Explore this through construction, and then justify your answer.
Yes, it is necessary to use the same radius for both arcs in a pair.
Justification: When constructing one pair of intersecting arcs (for example, both arcs above XY), the two arcs must intersect at a point that is equidistant from both X and Y. If different radii are used for the arc from X and the arc from Y, the intersection point will not be equidistant from X and Y. ∴ The point will not lie on the perpendicular bisector of XY.
Explanation
The chapter explains that for a point to lie on the perpendicular bisector of XY, it must be at equal distances from both X and Y. When we draw an arc from X with radius r₁ and an arc from Y with radius r₂, their intersection point A satisfies AX = r₁ and AY = r₂. For A to be equidistant from X and Y (and hence lie on the perpendicular bisector), we must have r₁ = r₂. If the radii are different, the intersection point will not be equidistant from X and Y, and will not lie on the perpendicular bisector.
Solution Steps
Recognize that the intersection point must be equidistant from both X and Y to lie on the perpendicular bisector.
Observe that using different radii creates an intersection point that is not equidistant from X and Y.
Conclude that equal radii are necessary for the construction to work correctly.