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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.
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.
ArgumentReason1Line segments AP, AQ, PB, QB are all congruentThe four distances were all drawn with the same compass width c.Next we prove that the top and bottom triangles are isosceles and congruent2Triangles APQ and BPQ are isoscelesTwo sides are congruent (length c)3Angles AQJ, APJ are congruentBase angles of isosceles triangles are congruent4Triangles APQ and BPQ are congruentThree sides congruent (sss). PQ is common to both.5Angles APJ, BPJ, AQJ, BQJ are congruent. (The four angles at P and Q with red dots)CPCTC. Corresponding parts of congruent triangles are congruentThen we prove that the left and right triangles are isosceles and congruent6APB and AQB are isoscelesTwo sides are congruent (length c)7Angles QAJ, QBJ are congruent.Base angles of isosceles triangles are congruent8Triangles APB and AQB are congruentThree sides congruent (sss). AB is common to both.9Angles PAJ, PBJ, QAJ, QBJ are congruent. (The four angles at A and B with blue dots)CPCTC. Corresponding parts of congruent triangles are congruentThen we prove that the four small triangles are congruent and finish the proof10Triangles APJ, BPJ, AQJ, BQJ are congruentTwo angles and included side (ASA)11The four angles at J - AJP, AJQ, BJP, BJQ are congruentCPCTC. Corresponding parts of congruent triangles are congruent12Each of the four angles at J are 90°. Therefore AB is perpendicular to PQThey are equal in measure and add to 360°13Line segments PJ and QJ are congruent. Therefore AB bisects PQ.From (8), CPCTC. Corresponding parts of congruent triangles are congruent
Try it yourselfClick 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.
Other constructions pages on this site
Circles, Arcs and Ellipses
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