Page 226 - ICSE Math 7

20 Constructions

Key Concepts

• Construction of Parallel Lines • Construction of Circumcircle and Incircle
• Construction of Triangles

Euclid, a Greek mathematician mentioned construction techniques in his book
titled Elements. These techniques have become a very important part of geometry
and are taught even today. He explained that construction can be done by using
only a compass and a ruler without measuring the lengths or angles.
In the previous class, we have learnt the construction of a circle, line segment,
perpendicular lines, perpendicular bisector of a line segment, special angles and
angle bisectors using a ruler and compass. In this lesson, we shall learn how to
draw parallel lines and triangles using a ruler and compass.

To Construct Parallel Lines

To draw a line parallel to the given line and passing through the given fixed point
Let AB be a line and P be the given fixed point. To draw a line parallel to P
AB passing through P, follow the steps given below.
A B
Step 1: Take any point Q on AB. Join P and Q. P

A Q B
Step 2: With Q as centre and a suitable radius, draw an arc intersecting P
QB at L and QP at M. M
A Q L B

Step 3: With P as centre and same radius, draw an arc on the opposite N P
side of PQ to intersect QP at N. M
A Q L B

O
Step 4: With N as centre, draw an arc of radius LM to intersect the arc N P
drawn in step 3 at O. M
A Q L B
O
Step 5: Join P and O and extend it. OP is the required line parallel to AB N P
and passing through P. M
A Q L B

We have constructed ∠MQL = ∠NPO because when a transversal cuts two parallel lines, then
the alternate angles so formed are equal. Thus, to construct a line parallel to a given line we have
drawn equal alternate angles.

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