Construction of Lines

Objective

This topic gives an overview of; 

  • Line Segment
  • Construction of a line segment of a given length
  • Constructing a copy of a given Line Segment.
  • Perpendicular to a line through a point on it
  • Perpendicular bisector of a line segment

A Line Segment

Remember that a line segment is bounded by two end-points. This makes it possible to measure its length with a ruler. If we know the length of a line segment, it becomes possible to represent it by a diagram. Let us see how we do this.

Construction of a line segment of a given length

Suppose we want to draw a line segment of length 4.7 cm. We can use our ruler and mark two points A and B which are 4.7 cm apart. Join A and B and get AB. While marking the points A and B, we should look straight down at the measuring device. Otherwise we will get an incorrect value.

Use of ruler and compasses

A better method would be to use compasses to construct a line segment of a given length.

Step 1: Draw a line l. Mark a point A on a line l.

Step 2: Place the compasses pointer on the zero mark of the ruler. Open it to place the pencil point upto the 4.7cm mark.

 

Step 3: Taking caution that the opening of the compasses has not changed, place the pointer on A and swing an arc to cut l at B.

Step 4: AB is a line segment of required length.

  

Constructing a copy of a given Line Segment.

Suppose you want to draw a line segment whose length is equal to that of a given line segment AB.

 A quick and natural approach is to use your ruler (which is marked with centimetres and millimetres) to measure the length of AB and then use the same length to draw another line segment CD

A second approach would be to use a transparent sheet and trace AB onto another portion of the paper. But these methods may not always give accurate results.

A better approach would be to use ruler and compasses for making this construction.

To make a copy of AB.

Step 1: Given AB whose length is not known.

Step 2: Fix the compasses pointer on A and the pencil end on B. The opening of the instrument now gives the length of AB

Step 3:  Draw any line l. Choose a point C on l. Without changing the compasses setting, place the pointer on C.

Step 4:  Swing an arc that cuts l at a point, say, D. Now CD is a copy of AB.

Perpendiculars

You know that two lines (or rays or segments) are said to be perpendicular if they intersect such that the angles formed between them are right angles.

In the figure, the lines l and m are perpendicular.

Perpendicular to a line through a point on it

Given a line l drawn on a paper sheet and a point P lying on the line. It is easy to have a perpendicular to l through P. We can simply fold the paper such that the lines on both sides of the fold overlap each other. Tracing paper or any transparent paper could be better for this activity. Let us take such a paper and draw any line l on it. Let us mark a point P anywhere on l. Fold the sheet such that l is reflected on itself; adjust the fold so that the crease passes through the marked point P. Open out; the crease is perpendicular to l.

Method of ruler and compasses

As is the preferred practice in Geometry, the dropping of a perpendicular can be achieved through the “ruler-compasses” construction as follows:

Step 1: Given a point P on a line l.    

Step 2:  With P as centre and a convenient radius, construct an arc intersecting the line l at two points A and B.

Step 3:  With A and B as centres and a radius greater than AP construct two arcs, which cut each other at Q.

Step 4: Join PQ. Then  is perpendicular to l. We write  ⊥ l.

Perpendicular to a line through a point not on it

If we are given a line l and a point P not lying on it and we want to draw a perpendicular to l through P, we can again do it by a simple paper folding as before. Take a sheet of paper (preferably transparent). Draw any line l on it. Mark a point P away from l. Fold the sheet such that the crease passes through P. The parts of the line l on both sides of the fold should overlap each other. Open out. The crease is perpendicular to l and passes through P.

The perpendicular bisector of a line segment

Fold a sheet of paper. Let AB be the fold. Place an ink-dot X, anywhere. Find the image X' of X, with AB as the mirror line.
Let AB and XX’ intersect at O. OX = OX'. This means that AB divides XX’ into two parts of equal length. AB bisects XX’ or AB is a bisector of XX’. Note also that ∠AOX and ∠BOX are right angles. Hence, AB is the perpendicular bisector of XX’. We see only a part of AB in the figure. The perpendicular bisector of a line joining two points is 
same as the axis of symmetry

Step 1: Draw a line segment AB.

Step 2: Place a strip of a transparent rectangular tape diagonally across AB with the edges of the tape on the end points A and B, as shown in the figure.
Step 3: Repeat the process by placing another tape over A and B just diagonally across the previous one. The two strips cross at M and N.
Step 4:  Join M and N.

Summary 

  •  Two lines are said to be perpendicular when they intersect each other at an angle of 90o.
  • The perpendicular bisector is a perpendicular line that bisects another line into two equal parts.
  • Using the ruler and compasses, the following constructions can be made:
    • A circle, when the length of its radius is known.
    • A line segment, if its length is given.
    • A copy of a line segment.
    • A perpendicular to a line through a point.
      • on the line
      • Not on the line.
    • The perpendicular bisector of a line segment of given length.

 

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