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# Perpendicular bisector of a line segment

This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This both bisects the segment (divides it into two equal parts, and is perpendicular to it. It finds the midpoint of the given line segment.

## Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

## Proof

This construction works by effectively building congruent triangles that result in right angles being formed at the midpoint of the line segment. The proof is surprisingly long for such a simple construction.

The image below is the final drawing above with the red lines and dots added to some angles. Argument Reason
1 Line segments AP, AQ, PB, QB are all congruent The four distances were all drawn with the same compass width c.
Next we prove that the top and bottom triangles are isosceles and congruent
2 Triangles ∆APQ and ∆BPQ are isosceles Two sides are congruent (length c)
3 Angles AQJ, APJ are congruent Base angles of isosceles triangles are congruent
4 Triangles ∆APQ and ∆BPQ are congruent Three sides congruent (sss). PQ is common to both.
5 Angles APJ, BPJ, AQJ, BQJ are congruent. (The four angles at P and Q with red dots) CPCTC. Corresponding parts of congruent triangles are congruent
Then we prove that the left and right triangles are isosceles and congruent
6 ∆APB and ∆AQB are isosceles Two sides are congruent (length c)
7 Angles QAJ, QBJ are congruent. Base angles of isosceles triangles are congruent
8 Triangles ∆APB and ∆AQB are congruent Three sides congruent (sss). AB is common to both.
9 Angles PAJ, PBJ, QAJ, QBJ are congruent. (The four angles at A and B with blue dots) CPCTC. Corresponding parts of congruent triangles are congruent
Then we prove that the four small triangles are congruent and finish the proof
10 Triangles ∆APJ, ∆BPJ, ∆AQJ, ∆BQJ are congruent Two angles and included side (ASA)
11 The four angles at J - AJP, AJQ, BJP, BJQ are congruent CPCTC. Corresponding parts of congruent triangles are congruent
12 Each of the four angles at J are 90°. Therefore AB is perpendicular to PQ They are equal in measure and add to 360°
13 Line segments PJ and QJ are congruent. Therefore AB bisects PQ. From (8), CPCTC. Corresponding parts of congruent triangles are congruent

- Q.E.D

## Try it yourself

Click here for a printable worksheet containing three bisection problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.