NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions

Question 75

Explanation (Step by Step, in Simple Words)

Goal: We want a line that cuts AB (length 7 cm) into two equal parts and is also at a right angle (90°) to AB. This special line is called the perpendicular bisector.

Why we draw two pairs of arcs?

Points that are the same distance from both ends A and B always lie on the perpendicular bisector of AB. So, we create two points P and Q that are equally far from A and B by drawing arcs with the same radius from A and from B.

What does “same distance” mean here?

This happens because we used equal radii for the arcs from A and from B.

Why does line PQ bisect AB?

Since P and Q are both the same distance from A and B, they must lie on the locus of points equidistant from A and B — that locus is exactly the perpendicular bisector of AB. So joining P and Q gives that line.

Why is it perpendicular?

The perpendicular bisector is at a right angle to the segment it bisects. Therefore,

Why does it “bisect” (cut into two equal halves)?

Let M be the point where PQ meets AB. Because M lies on the perpendicular bisector,

Since AB is 7 cm, each half is

Quick checklist (to be sure you did it right):

• P and Q are the intersection points of the arcs (one pair above and one pair below AB).
• Line PQ cuts AB at M such that

  ( AM = MB )

  .
• PQ makes a right angle with AB (90°):

  ( PQ perp AB )

  .

Small idea to remember:

“Equal arcs from the two ends → intersection points are equally distant from both ends → join those points → you get the perpendicular bisector.”