This construction shows how to draw the perpendicular bisectorof a given line segmentwith compass and straightedge or ruler. This both bisects the segment (divides it into two same parts, and also is perpendicularto it. It find the midpoint the the offered line segment.

## Printable step-by-step instructions

The above animation is accessible as a printable step-by-step accuse sheet, which deserve to be used for make handoutsor once a computer system is no available.

You are watching: When constructing a perpendicular bisector why must the compass

## Proof

This building works by effectively building congruent triangle that result in best angles beingformed in ~ the midpoint that the line segment. The evidence is surprisingly long for such a basic construction.

The image below is the last drawing over with the red lines and also dots included to part angles.

ArgumentReason1Line segment AP, AQ, PB, QB room all congruentThe four ranges were all drawn with the very same compass broad c.Next we prove that the top and bottom triangles space isosceles and also congruent2Triangles ∆APQ and also ∆BPQ are isoscelesTwo sides room congruent (length c)3Angles AQJ, APJ space congruentBase angles of isosceles triangles space congruent4Triangles ∆APQ and also ∆BPQ space congruentThree sides congruent (sss). PQ is usual to both.5Angles APJ, BPJ, AQJ, BQJ are congruent.(The four angles in ~ P and Q v red dots)CPCTC. Matching parts that congruent triangles room congruentThen us prove the the left and also right triangles space isosceles and also congruent6∆APB and also ∆AQB are isoscelesTwo sides room congruent (length c)7Angles QAJ, QBJ are congruent.Base angle of isoscelestriangles space congruent8Triangles ∆APB and also ∆AQB are congruentThree sides congruent (sss). Abdominal muscle is usual to both.9Angles PAJ, PBJ, QAJ, QBJ are congruent.(The four angles at A and B through blue dots)CPCTC. Corresponding parts that congruent triangles room congruentThen we prove the the four tiny triangles room congruent and finish the proof10Triangles ∆APJ, ∆BPJ, ∆AQJ, ∆BQJ room congruentTwo angles and included next (ASA)11The four angles in ~ J - AJP, AJQ, BJP, BJQ are congruentCPCTC. Matching parts the congruent triangles room congruent12Each of the four angles in ~ J room 90°. Therefore abdominal muscle is perpendicular come PQThey space equal in measure and include to 360°13Line segments PJ and QJ are congruent. Therefore abdominal bisects PQ.From (8), CPCTC. Corresponding parts that congruent triangles are congruent-Q.E.D

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