# 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. 