Practical Geometry: Construction of Lines, Angles and Circles

Take in the beauty of your surroundings. Numerous objects of all forms and sizes can be found, and we even design several pictures that include other figures. Practical Geometry entails learning about drawing tools such as a scale, protractor, and a pair of compasses, as well as how to draw lines, line segments, perpendicular lines, angles, and circles using these tools.

With the use of drawing tools such as a scale, protractor, and a set of compasses, we will learn to create preliminary figures of geometry such as lines, circles, angles, and so on. Continue reading to know more.

Introduction to Geometry

Geometry is the study of the position, shape, size, and other properties of different figures. Geometrical figures such as point, line, plane, etc., carry the basic idea for the development of geometry. 

Let us look at the terminology we will use throughout our blog and the basic foundation of geometry:

1. Point: A point is a mark of position. It has neither length nor width nor thickness and occupies no space. A dot can represent a point. 
2. Line: A line has the only length. It has neither width nor thickness.  
3. Ray: It is a straight line that starts from a fixed given point and moves in the same direction.
4. Line Segment: A line segment is a part of a straight line as well as a ray.
5. Surface: A surface has length and width but no thickness.
6. Plane: It is a flat surface. A plane has length and width but no thickness.

Tools Used for Construction

To construct the shapes, we need some tools. Let’s have a look at the tools, describing them and how to use them.

1. Scale or a Ruler: Normally, a ruler has no marking, but a ruler in our geometry box is graduated with centimetres along one edge and sometimes into inches along the other edge. A scale helps us to draw line segments and to measure their lengths.
2. A Pair of Compasses: A compass is an instrument used for drawing circles or marking measurements consisting of two pointed legs, a pointer on one end, and a pencil on the other. 
3. Divider: A divider consists of a pair of legs, sharply pointed at both ends. A divider helps us to measure the length of the line segments. 
4. Set Squares: A set square consists of two triangular pieces. One of them has \({45^{\rm{o}}},{45^{\rm{o}}},{90^{\rm{o}}}\) angles at the vertices, and the other has \({30^{\rm{o}}},{60^{\rm{o}}},{90^{\rm{o}}}\) angles at the vertices. A set square helps us to draw perpendicular and parallel lines.
5. Protractor: A semi-circular device graduated into \({180^{\rm{o}}}\) parts. The measure starts from \({0^{\rm{o}}}\) on the right-hand side and ends with \({180^{\rm{o}}}\) on the left-hand side and vice versa. A protractor helps us to draw and measure angles.

What is a Circle?

A circle is a closed curve such that all the points on its circumference or boundary are equidistant from a fixed point inside it. Circles are round, two-dimensional-shaped figures. All points on the circle’s boundary are at an equal distance from a point called the centre. The radius of a circle is a line segment connecting the centre of the circle to any point on the circle’s boundary.

The fixed point is the centre of the circle, and it is denoted by \(O;\) the constant distance \(r\) from the centre to any point on the boundary corresponds to the circle’s radius.

Construction of Circle

Let us draw a circle of radius \({\rm{4}}\,{\rm{cm}}{\rm{.}}\) For this, we need to use compasses. Below are the steps to be followed:

1. Open the compasses for the required radius of \({\rm{4}}\,{\rm{cm}}{\rm{.}}\)

2. Mark a point with a sharp pencil where we want the centre of the circle to be. Name the centre of the circle as \(O.\)

3. Place the pointer of the compasses on \(O.\)

4. Turn the compass slowly to draw the circle and complete the move around in one instant.

What is a Line Segment? 

A part of a line is called a line segment. It has two endpoints and has a definite length, which cannot be increased or decreased.

Line segment \(AB\) can be written as \(\overline {AB} .\) Between any two points, we can draw only one line segment. Physically an edge of a table, the edge of a ruler, etc., represents line segments.

Construction of Line Segment

Suppose to draw a line segment of \(7.3\,{\rm{cm,}}\) and we can use our ruler and mark two points A and B which are \(7.3\,{\rm{cm}}\) apart. Join \(A\) and \(B\) and get \(\overline {AB} .\) While marking, we should look straight down at the measuring device, or we will get an incorrect value.

Construction of Line Segment Using Ruler and Compasses

We can draw a line segment with the help of a ruler and a compass.

Draw a line \(l.\) Mark the point \(A\) on the line.

Place the compasses pointer on the \(0\) marks of the ruler and open it to place the pencil point up to the \(7.3\,{\rm{cm}}\) mark.

Make sure that the opening of the compasses has not changed, place the pointer on \(A\) and swing an arc to cut \(l\) at \(B.\)

\(\overline {AB} \) is a line segment of the required length.

Perpendicular to a Line Segment

Closely look at the pair of set squares. Each set square has three edges. The two sides of each set square contain an angle of \({90^{\rm{o}}};\) hence these two sides are perpendicular to each other. Thus, two lines are said to be perpendicular to each other if they contain an angle of \({90^{\rm{o}}}\) between them.

Construction of Perpendicular Lines Using Ruler and Compass

Mark a point \(P\) on the line \(l.\)

With \(P\) as a centre and by taking the convenient radius, construct an arc intersecting the line \(l\) at two points \(A\) and \(B.\)

By taking \(A\) and \(B\) as centres and radius greater than \(AP\) construct two arcs, which cut each other at \(Q.\)

Join \(PQ \cdot \mathop {PQ}\limits^ \leftrightarrow \) is perpendicular to \(l.\)

Angles

Let us consider the rotation of \(\overrightarrow {OB} \) around its initial point \(O.\) It takes a new position \(\overrightarrow {OA} \) as shown below in the figure.

Here, \(\overrightarrow {OA} \) and \(\overrightarrow {OB} \) are said to form an angle. The two different rays starting from the same fixed point form an angle. The common initial point of the two rays forming the angle is called the vertex of the angle.

Constructing Angles

Let us learn to construct an angle of the desired measurements using a protractor.

Construct \(∠XYZ\) of \({65^{\rm{o}}}\) with the help of a protractor.Draw a ray \(YZ.\)

Place a protractor in such a way that its centre lies on point \(Y\) and its baseline coincides with ray \(YZ.\)

Mark a point \(X\) against the \({65^{\rm{o}}}\) mark on the inner scale.

Remove the protractor and draw ray \(YX.\)

\(∠XYZ\) is the required angle.

Solved Examples

Q.1. In figure line, segments \(PQ\) and \(RQ\) are marked on a line \(l\) such that \(PQ=AB\) and \(RQ=CD.\) Find the measure of \(AB-CD.\)

Ans: From the figure, we can see that

\(PQ=AB\) and \(RQ=CD\)
\(AB-CD=PQ-RQ=PR\)

Hence, \(AB-CD=PR\)

Q.2. In the given figure, if \(CD = 20\;{\rm{cm}}\) and \(A\) and \(B\) are centres of the circle, then find the measure of \(MD\) if the line joining the midpoints divides \(CD\) into two equal parts.

Ans: In the given figure,

The line \(AB\) joining the centres of the circles bisects the line \(CD\) into \(2\) equal parts.
Given that the line \(CD = 20\;{\rm{cm}}\)
Therefore, \(MD = \frac{{CD}}{2} = \frac{{20}}{2}\;{\rm{cm}} = 10\;{\rm{cm}}\)
Hence, the measure of \(MD = 10\;{\rm{cm}}.\)

Q.3.In the given figure, find out the number of angles within the arms \(OA\) and \(OE.\)

Ans: The angles formed within the arms of \(OA\) and \(OE\) are \(\angle EOD,\angle EOC,\angle EOB,\angle EOA,\angle DOC,\angle DOB,\angle DOA,\angle COB,\angle COA,\angle BOA.\)Hence \(10\) angles are formed within the arms \(OA\) and \(OE.\)

Q.4. For the figure given below, write the name of the vertex, the name of the arms, and the name of the angle.

Ans: Arms \(=AB\) and \(BC,\) vertex \(=O\) and angle\(=∠ABC\)

Q.5. Define set square and mention the angles of each of the set squares.
Ans: A set square consists of two triangular pieces. One of them has \({45^{\rm{o}}},{45^{\rm{o}}},{90^{\rm{o}}}\) angles at the vertices, and the other has \({30^{\rm{o}}},{60^{\rm{o}}},{90^{\rm{o}}}\) angles at the vertices.

Summary

In this article, we learned about the basics of geometry. We learned about the tools used in drawing and constructing geometrical figures. Later with the help of tools like compasses, rulers, and protractors, we learned to draw line segments, perpendicular lines, angles, and circles.

Frequently Asked Questions

We have provided some frequently asked questions about Practical Geometry here:

Q.1. What is practical geometry?
Ans: Practical geometry is an essential branch of geometry that deals with studying the shape, size, and dimensions of objects.

Q.2. What are the three tools of geometry?
Ans: The three main tools of geometry are a ruler, compasses, and a protractor.

Q.3. Who is the father of geometry?
Ans: Euclid is considered the father of geometry.

Q.4. How do you draw angles in practical geometry?
Ans: We draw angles with the help of a compass and a protractor. 

Q.5. Are angles part of geometry?
Ans: Yes, angles are a part of geometry.

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