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.

Question

Bright Tutorials · Accepted Answer

Draw a line segment XY.
With centres at X and Y, draw arcs of unequal radii, say k and k’.
Let the arcs intersect at the point A.
Let PQ be the perpendicular bisector of XY.
Let R be any point on PQ.
Join RX and RY.
∆ROX and ∆ROY are congruent, because OX = OY, ∠ROX = ∠ROY, and OR is common.
∴ RX = RY
∴ For any point R on the perpendicular bisector, we have RX = RY.
∴ Every point on the perpendicular bisector is equidistant from X and Y.
Since AX ≠ AY, the point A is not on the perpendicular bisector.
Thus, to construct a perpendicular bisector, we must use the same radius for both arcs of a pair of arcs.

Constructions and Tilings — Question 3