PERPENDICULAR BISECTOR OF A LINE SEGMENT

Bisection Of Line Segment: Perpendicular Bisector Of A Line Segment

Line Segment Bisector is a line, ray or segment which cuts another line segment into two equal parts. If you move one of the orange dots at A or B, you would observe that the line AB at all times divides the line PQ into two equal parts.

Generally, ‘to bisect’ something means to cut it into two equal parts. The ‘bisector’ is what does the cutting.

With a line bisector, we are cutting a line segment into two equal lengths with another line – the bisector. In the figure above, the line PQ was cut into two equal lengths (PF and FQ) by the bisector line AB.

If AB crosses at a right angle, it is known as the “perpendicular bisector” of PQ. If it crosses at any other angle it is merely known as a bisector. For reasons that are clear, the point F is known as the midpoint of the line PQ.

How to bisect a line segment

Clearly, one way to bisect a line segment is to measure its length, divide that by two and mark the midpoint.
But you can do it without any measurement at all with the use of a mere compass and straightedge through techniques developed thousands of years ago by the Greeks.

Perpendicular bisector of a line segment

This construction below shows the way to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This perpendicular bisector both bisects the segment (divides it into two equal parts, and is as well perpendicular to the segment. It gives the midpoint of the given line segment.

Step-by-step instructions on how perpendicular bisector of a line segment works

This step by step instruction sheet can be very handy for making lesson note as well for learning on your own but in the presence of the computer and in the absence of it.

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 picture below is how the final drawing will look like. It has some red lines and dots added to a few 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.
	After that, we prove that the top and bottom triangles are isosceles and congruent
2	Triangles ∆APQ and ∆BPQ are isosceles	Two sides are congruent ( all with lengths as 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 indicated with red dots)	CPCTC. Corresponding parts of congruent triangles are congruent
	At this point 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, QBJ 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. (These are indicated by four angles at A and B with blue dots)	CPCTC. Corresponding parts of congruent triangles are congruent
	After that, we prove that the four small triangles are congruent and conclude 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. Equivalent parts of congruent triangles are congruent
12	Each of the four angles at J are 90°. Thus AB is perpendicular to PQ	They are equal in measure and add to 360°
13	Line segments PJ and QJ are congruent. Thus AB bisects PQ.	From (8), CPCTC. Equivalent parts of congruent triangles are congruent

– Q.E.D

Construction of the perpendicular bisector of a line segment

This is the step-by-step, printable instruction to guide you when working on your own with or without computer.

	After you have done this	Your work ought to look like this
	Begin with a line segment PQ.
1	Place the compasses on one end of the line segment.
2	Set the compasses’ width to a roughly two thirds the line length. The actual width does not matter.
3	Without altering the compasses’ width, draw an arc above and below the line.
4	Again without altering the compasses’ width, place the compasses’ point on the other end of the line. Draw an arc above and below the line in order that the arcs cross the first two.
5	Making use of a straightedge, draw a line between the points where the arcs intersect.
6	Done. This line is perpendicular to the first line and bisects it (cuts it at the exact midpoint of the line).

Bisect means to cut into two congruent (equal) pieces. To bisect a line segment is also known as to Construct a Perpendicular Bisector of a segment.

Given: (Line segment) Question: Bisect .

Directions

Bisect an angle

Given: Question: Bisect .

Directions:

1. Place the point of the compass on the vertex of (point A). 2. Stretch the compass to any length very long as it stays ON the angle. 3. Swing an arc to allow the pencil to cross both sides of . This will produce two intersection points with the sides (rays) of the angle. 4. Put the compass point on one of these new intersection points on the sides of .

If required, stretch your compass to an enough length to place your pencil well into the interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this interior – you do not require crossing the sides of the angle.
5. Without altering the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc. Your two small arcs in the interior of the angle ought to be crossing.
6. Attach the point where the two small arcs cross to the vertex A of the angle.
You have now produced two new angles that are of equal length (and are each 1/2 the measure of .)

Explanation of construction:

To comprehend the explanation, a few additional labeling will be required. Label the point where the arc crosses side as D. Label the point where the arc crosses side as E. And label the intersection of the two small arcs in the interior as F. Draw segments and . By the construction, AD = AE (radii of same circle) and DF = EF (arcs of equal length). Of course AF = AF. All of these groups of equal length segments are as well congruent. We have congruent triangles by SSS. De to the fact that the triangles are congruent, any of their leftover equivalent parts are congruent which makes equal (or congruent) to .

1. Place your compass point on A and stretch the compass MORE THAN half way to point B, but not beyond B. 2. With this length, swing a large arc that will go BOTH above and below . (If you do not wish to make one large continuous arc, you may simply place one small arc above and one small arc below .) 3. Without changing the span on the compass, place the compass point on B and swing the arc again. The two arcs you have created should intersect.

4. With your straightedge, connect the two points of intersection. 5. This new straight line bisects . Label the point where the new line and cross as C. has now been bisected and AC = CB. (It could also be said that the segments are congruent, .)

(It may be advantageous to instruct students in the use of the “large arc method” because it creates a “crayfish” looking creature which students easily remember and which reinforces the circle concept needed in the explanation of the construction.)

Explanation of construction: To understand the explanation you will need to label the point of intersection of the arcs above segment as D and below segment as E. Draw segments , , and . All four of these segments are of the same length since they are radii of two congruent circles. More particularly, DA = DB and EA = EB.

Now, bear in mind of a locus theorem: The locus of points equidistant from two points, is the perpendicular bisector of the line segment determined by the two points. Thus, is the perpendicular bisector of . The fact that the bisector is as well perpendicular to the segment is in fact MORE than we required for a simple “bisect” construction.

Line Segment

A line segment is a straight line which links two points without extending beyond them.

If you alter the line segment below by dragging an orange dot on an endpoint and would observe how the segment PQ acts.

The line segment PQ links the points P and Q. The points P and Q are known as the ‘endpoints’ of the segment. The word ‘segment’ basically means ‘a piece’ of something, and here it means the piece of a full line, which would usually extend to infinity in both directions.

A line segment is one-dimensional. It is made up of a measurable length, but has zero width. If you draw a line segment with a pencil, examination with a microscope would illustrate that the pencil mark has a measurable width. The pencil line is merely a way to show the idea on paper. In geometry, nevertheless, a line segment has no width.

Naming of line segments

Line segments are basically named in two ways:

1. By the endpoints.
In the figure above, the line segment would be called PQ because it links the two points P and Q. Recall that points are normally tagged with single upper-case (capital) letters. A shorthand way of writing this is /PQ/. This is read as “line segment PQ”. The bar over the two letters indicates it is a line segment,rather than a line, which goes on forever in both directions.

2. By a single letter.
The segment above would be called simply “y”. By convention, this is usually a single lower case (small) letter. This method is often used in the naming the sides of triangles and other polygons.

Construction of perpendicular at a point on a line

This is the step-by-step instruction on how to construct perpendicular at a point on a line.

	After doing this	Your work ought to look like this
	Begin with a line and point K on that line.
1.	Set the compasses’ width to a medium setting. The actual width does not matter.
2	Without altering the compasses’ width, mark a short arc on the line at each side of the point K, forming the points P, Q. These two points are therefore the same distance from K.
3	Raise the compasses to almost double the width (again the exact setting is not significant).
4	From P, mark off a short arc above K
5	Without changing the compasses’ width repeat from the point Q so that the two arcs cross each other,producing the point R
6	Making use of the straight edge, draw a line from K to where the arcs cross.
7	Done. The line merely drawn is a perpendicular to the line at K