Chapter 14 - Practical Geometry

Introduction to Practical Geometry

Starting from basic objects like tables and books to the world-famous monuments, everything can be defined in terms of their geometrical shapes.

The Geometry Box

We can draw basic geometrical shapes like lines, triangles, rectangles, and circles with the help of the instruments available in a geometry box.

Ruler − Use this instrument to draw uniform line segments.
Divider − Use this instrument to compare the length of two line segments.
Compass − To draw an arc or a circle
Set Squares − To draw triangle or perpendicular lines
Protractor − To measure angles
Others (Pencil, Eraser, Sharpener) − Use these instruments for drawing purpose.

Geometrical ideas are reflected in all forms of art, measurements, architecture, engineering, and designing.

Construction of Line Segments and Circles

A ruler can be used to construct a line segment and a compass can be used to construct a circle.

Drawing a Line Segment

To draw a line segment using a ruler,

Consider a length of our choice, say 10 cm.
Take the ruler and draw a line segment starting from 0 cm to 10 cm.

We can also use the ruler to measure and compare the length of different line segments.

Comparing Two Line Segments

We use the divider to compare the length of different line segments.

Fix the tip of the legs of divider on the ends of the line segment.
The more the gap between the two legs, the longer the line.

It is also possible to measure the length of a line segment using a divider along with a ruler.

Drawing a Circle using Compass

Let's fix the radius of the circle as 4 cm.

Take a ruler and stretch the arms of the compass such that one arm is fixed at 0 and the other arm at 4 cm.
Hold the compass on the top and fix the other arm of compass firmly on the paper and rotate the pencil arm.

In this way, we draw a perfect circle of 4 cm radius.

Construction of Perpendicular Lines

The line which is making an angle 90° to another line is known as the perpendicular to the line. Perpendicular lines can be drawn using set squares or a compass.

There are four ways to draw perpendicular lines.

Method 1 − Using Ruler and Set Square

Draw a line c and mark a point A on it.
Place a ruler with one of its edges along l.
Place a set square on the line such that the 90° facing edge coincides with the line.
Slide the set square along the edge of ruler until its right angled corner coincides with A.
Draw a desired line where the set square coincides with A and name its end-point as B.
The new line AB is perpendicular to the line c.

Method 2 − Using Ruler and Set Square

Draw a straight line l and mark a point P outside it.
Place a set square on l such that one arm of its right angle aligns along l.
Place a ruler along the edge opposite to the right angle of the set square.
Hold the ruler fixed and slide the set square along the ruler till the point P.
Stop sliding when the right angle touches the point P.
Draw a line segment joining the point P towards the line l.

Method 3 − Using Ruler and Compass

Draw a line l and mark a point P on it.
Take the compass and select a convenient radius.
With P as centre, construct an arc using the compass.
The arc should intersect the line l at two points A and B.
With A and B as centres and a radius greater than AP construct two arcs, which cut each other at Q.
Join PQ to get the desired perpendicular.

Method 4 − Using Ruler and Compass

Draw a line l and mark a point P outside it.
With P as centre, draw an arc which intersects line l at two points A and B.
Using the same radius and with A and B as centres, construct two more arcs that intersect at a point, say Q, on the other side.
Join PQ to get the desired perpendicular.

Construction of Perpendicular Bisectors

The word 'bisect' means dividing into two equal parts.

If we use a perpendicular line to divide a line into two equal parts, then that perpendicular line is called a perpendicular bisector.

Drawing a Perpendicular Bisector

Draw a line segment AB of any length.
With A and B as the centres, draw four arcs with the compass on both sides of AB such that they intersect at two points.
Name these two points as P and Q.
The radius should be more than half the length of AB.
Join the points P and Q. It cuts AB at O.
PQ is the perpendicular bisector of AB.

In the above figure,

AO = OB = ${1}/{2}$AB

You can use a ruler to verify that AO = OB.

Construction of Angles

A shape, formed by two lines or rays diverging from a common point (the vertex) is known as an angle.

Angles can be of different measures like 40°, 60°, 90°, 180°, etc.

Drawing an Angle using Protractor

Suppose we want to draw an acute angle of 40°.

We use the protractor to draw an angle.

The steps are as follows:

Draw a line segment AB of any length.
Take the protractor and place its centre on the point A.
The 0° mark of the protractor should overlap with the line segment AB.
Now, mark 40° using the protractor. Name this point as C.
As the line segment falls on the right side of the centre, mark the 40° that is to the right.
Join the points A and C.
∠BAC is the required angle.

Had the line segment been to the left of the centre, then we would have marked the 40° that is on the left of the protractor.

Construction of Angle Bisectors

Angle bisector is the line that divides an angle into two equal halves. For example, the bisector of a 60° angle will produce two angles of 30° each.

Drawing an Angle Bisector

The steps are as follows:

Suppose we have an angle ∠A.
Take the compass with a convenient radius of your choice and draw an arc with A as the centre
The arc should cut both the arms of ∠A.
Label the points of intersection as P and Q.

Using the same radius, from P and Q as centres, draw two arcs such that they meet at a point (in the interior of ∠A).

Name the intersecting point as S.
Join the points A and S.
The line segment AS is the angle bisector of ∠A.

AS bisects the angle ∠A into two equal halves.