This building reflects how to draw the perpendicular bisectorof a given line segmentthrough compass and straightedge or leader. This both bisects the segment (divides it into two equal components, and is perpendicularto it. It finds the midallude of the given line segment.

Printable step-by-step instructions

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

Proof

This building and construction works by properly building congruent triangles that cause best angles beingformed at the midsuggest of the line segment. The proof is surprisingly long for such a simple building.

The picture listed below is the last illustration over through the red lines and also dots included to some angles.


ArgumentReason1Line segments AP, AQ, PB, QB are all congruentThe four distances were all attracted through the very same compass width c.Next we prove that the top and also bottom triangles are isosceles and also congruent2Triangles ∆APQ and also ∆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 widespread to both.5Angles APJ, BPJ, AQJ, BQJ are congruent.(The 4 angles at P and Q via red dots)CPCTC. Corresponding parts of congruent triangles are congruentThen we prove that the left and also right triangles are isosceles and also congruent6∆APB and ∆AQB are isoscelesTwo sides are congruent (length c)7Angles QAJ, QBJ are congruent.Base angles of isoscelestriangles are congruent8Triangles ∆APB and ∆AQB are congruentThree sides congruent (sss). AB is common to both.9Angles PAJ, PBJ, QAJ, QBJ are congruent.(The 4 angles at A and B via blue dots)CPCTC. Corresponding parts of congruent triangles are congruentThen we prove that the 4 tiny triangles are congruent and complete the proof10Triangles ∆APJ, ∆BPJ, ∆AQJ, ∆BQJ are congruentTwo angles and also contained side (ASA)11The four angles at J - AJP, AJQ, BJP, BJQ are congruentCPCTC. Corresponding components of congruent triangles are congruent12Each of the four angles at J are 90°. Therefore AB is perpendicular to PQThey are equal in meacertain and add to 360°13Line segments PJ and also QJ are congruent. Because of this AB bisects PQ.From (8), CPCTC. Corresponding components of congruent triangles are congruent-Q.E.D


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