Chapter 14 - Practical Geometry

Introduction to Practical Geometry

Starting from basic objects pke 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 pke pnes, triangles, rectangles, and circles with the help of the instruments available in a geometry box.

Ruler − Use this instrument to draw uniform pne segments.

Divider − Use this instrument to compare the length of two pne segments.

Compass − To draw an arc or a circle

Set Squares − To draw triangle or perpendicular pnes

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 pne segment and a compass can be used to construct a circle.

Drawing a Line Segment

To draw a pne segment using a ruler,

Consider a length of our choice, say 10 cm.

Take the ruler and draw a pne segment starting from 0 cm to 10 cm.

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

Comparing Two Line Segments

We use the spanider to compare the length of different pne segments.

Fix the tip of the legs of spanider on the ends of the pne segment.

The more the gap between the two legs, the longer the pne.

It is also possible to measure the length of a pne segment using a spanider 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 pne which is making an angle 90° to another pne is known as the perpendicular to the pne. Perpendicular pnes can be drawn using set squares or a compass.

There are four ways to draw perpendicular pnes.

Method 1 − Using Ruler and Set Square

Draw a pne c and mark a point A on it.

Place a ruler with one of its edges along l.

Place a set square on the pne such that the 90° facing edge coincides with the pne.

Spde the set square along the edge of ruler until its right angled corner coincides with A.

Draw a desired pne where the set square coincides with A and name its end-point as B.

The new pne AB is perpendicular to the pne c.

Method 2 − Using Ruler and Set Square

Draw a straight pne l and mark a point P outside it.

Place a set square on l such that one arm of its right angle apgns along l.

Place a ruler along the edge opposite to the right angle of the set square.

Hold the ruler fixed and spde the set square along the ruler till the point P.

Stop spding when the right angle touches the point P.

Draw a pne segment joining the point P towards the pne l.

Method 3 − Using Ruler and Compass

Draw a pne 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 pne 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 pne l and mark a point P outside it.

With P as centre, draw an arc which intersects pne 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 spaniding into two equal parts.

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

Drawing a Perpendicular Bisector

Draw a pne 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 pnes or rays spanerging from a common point (the vertex) is known as an angle.

Angles can be of different measures pke 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 pne 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 pne segment AB.

Now, mark 40° using the protractor. Name this point as C.

As the pne 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 pne 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 pne that spanides 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 pne segment AS is the angle bisector of ∠A.

AS bisects the angle ∠A into two equal halves.