Constructions and Tilings — Question 2

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’ above XY. With centres at A and Y, draw arcs of radius ‘k’’ above XY. Let the arcs intersect at the points A and B. Join A and B and produce this line to 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, the pairs of arcs are both on the same side of XY. ∴ It is not necessary to construct the pairs of arcs above and below XY.