When copying a segment and bisecting a segment which step is the same

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.

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
-Q.E.D

Try it yourself

Other constructions pages on this site

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of n line segments
• Difference of two line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
• Perpendicular from a line through a point
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy)
• Parallel line through a point (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Copy an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a 60° angle
• Construct a 90° angle (right angle)
• Sum of n angles
• Difference of two angles
• Supplementary angle
• Complementary angle
• Constructing 75° 105° 120° 135° 150° angles and more

Triangles

• Copy a triangle
• Isosceles triangle, given base and side
• Isosceles triangle, given base and altitude
• Isosceles triangle, given leg and apex angle
• Equilateral triangle
• 30-60-90 triangle, given the hypotenuse
• Triangle, given 3 sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two angles and non-included side (aas)
• Triangle, given two sides and included angle (sas)
• Triangle medians
• Triangle midsegment
• Triangle altitude
• Triangle altitude (outside case)

Right triangles

• Right Triangle, given one leg and hypotenuse (HL)
• Right Triangle, given both legs (LL)
• Right Triangle, given hypotenuse and one angle (HA)
• Right Triangle, given one leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Tangents to two circles (external)
• Tangents to two circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given one side
• Square inscribed in a circle
• Hexagon given one side
• Hexagon inscribed in a given circle
• Pentagon inscribed in a given circle

Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object