Constructions and Tilings — Question 1

Draw a line segment XY. Choose distances k and k’ which are slightly greater than half of the distance XY. With centres at X and Y, draw arcs of radius ‘k’ below XY. With centres at X and Y, draw arcs of radius ‘k’’ below XY. Let the arcs above XY intersect at A, and the arcs below XY intersect at B. Join A and B. Let AB intersect XY at O. Join AX, AY, BX, and BY. ∆ABX and ∆ABY are congruent because AX = AY = k, BX = BY = k’, and AB is common. ∴ ∠XAO = ∠YAO ∆AOX and ∆AOY are congruent because AX = AY = k, ∠XAO = ∠YAO, and OA is common. ∴ OX = OY and ∠AOX = ∠AOY Also, ∠AOX + ∠AOY = 180° ∴ 2∠AOX = 180° or ∠AOX = 90° ∴ OX = OY and ∠AOX = ∠AOY = 90°. ∴ AB is the perpendicular bisector of the line XY. Here, A and B are points that are of the same distance from X and Y. Thus, any point that is of the same distance from X and Y lies on the perpendicular bisector.