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PART - 2 CH : 6 Constructions and Tilings — MATHS STD 7 — Question

Gujarat BoardEnglish MediumSTD 7MATHSPART - 2 CH : 6 Constructions and Tilings5 Marks
Question
When constructing the perpendicular bisector, is it necessary to have the same radius for the arcs above and below XY? Explore this through construction, and then justify your answer.
[Hint 1: Any point that is of the same distance from X and Y lies on the perpendicular bisector.
Hint 2: We can draw the whole line if any two of its points are known.]
✓

Answer

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.
Image
∆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.

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