Question
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.
✓
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.
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.
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