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Perpendicular bisector of a line segment
Geometry construction using a compass and straightedge
Step-by-step Instructions Printer friendly version
After doing this Your work should look like this
Start with a line segment PQ. Geometry construction with compass and straightedge or ruler or ruler
1.   Place the compass on one end of the line segment. Geometry construction with compass and straightedge or ruler or ruler
2.   Set the compass width to a approximately two thirds the line length. The actual width does not matter. Geometry construction with compass and straightedge or ruler or ruler
3.   Without changing the compass width, draw an arc on each side of the line. Geometry construction with compass and straightedge or ruler or ruler
4.   Again without changing the compass width, place the compass point on the the other end of the line. Draw an arc on each side of the line so that the arcs cross the first two. Geometry construction with compass and straightedge or ruler or ruler
5.   Using a straightedge, draw a line between the points where the arcs intersect. Geometry construction with compass and straightedge or ruler or ruler
6.   Done. This line is perpendicular to the first line and bisects it (cuts it at the exact midpoint of the line). Geometry construction with compass and straightedge or ruler or ruler

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, PAJ 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 line 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.

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(C) 2009 Copyright John Page