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Drawing the Perpendicular Bisector of a Line Segment

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The perpendicular bisector of a line segment:

Fold a sheet of paper. Let `bar"AB"` be the fold. Place an ink-dot X, as shown, anywhere.

Find the image X' of X, with AB as the mirror line.

Let `bar"AB"` and `bar"XX’"` intersect at O.

This means that `bar"AB"  "divides"  bar"XX’"` into two parts of equal length.

`bar"AB"  "bisects"  bar"XX’" or bar"AB"  "is a bisector of"  bar"XX’"`.

Hence, `bar"AB"  "is the perpendicular bisector of"  bar"XX’"`.

Construction using ruler and compasses:

Step 1: Draw a line segment `bar"AB"` of any length.

Step 2: With A as the centre, using compasses, draw a circle. The radius of your circle should be more than half the length of `bar"AB"`.

Step 3: With the same radius and with B as centre, draw another circle using compasses. Let it cut the previous circle at C and D.

Step 4: Join `bar"CD". "It cuts"  bar"AB"` at O.

Use your divider to verify that O is the midpoint of `bar"AB"`.

Therefore, `bar"CD"  "is the perpendicular bisector of"   bar"AB".`

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