Chapter 14

Practical Geometry

All kinds of shapes surround us. Each figure is composed of straight or curved lines of varying lengths and inclined at different angles. Hence, we must draw them using a set of geometrical tools, such as:

    1. Ruler: The ruler's straight edge can be used to draw line segments of various lengths.

    2. Compass: This is used to mark off equal lengths without measuring them.

    3. Divider: This is used to compare lengths.

    4. Set-Squares: This is used to draw perpendicular and parallel lines.

    5. Protractor: This is used to draw and measure angles.

 

In this chapter, we will learn to draw different geometrical figures using the ruler and compass mostly. To achieve better results:

    • Draw thin lines and mark points lightly.

    • Maintain instruments with sharp tips and fine edges.

 

In this chapter, we shall discuss:

    1. Circle

    2. Line Segment

    3. To construct a copy of a given line segment

    4. Perpendicular segments

    • Perpendicular to a line through a point on it (using a compass and a scale)

    • Perpendicular to a line through a point not on it (using a compass and ruler)

    • Perpendicular bisector of a line segment (using a compass and ruler)

 

5. Angles

    • Angles of specific measures

    • Angle bisector

    • Construction of a copy of an angle of unknown measure

 

6. Some angles of special measures

Circle

The circle is perhaps one of the most common shapes around us. Every point on a circle is at the same distance from the centre of the circle. That distance is called its radius. Examples of a circle include a wheel, ring, bangle, etc.

To construct a circle of a given radius using a compass, follow these steps:

    1. Open the compass along the scale to measure a radius of x cm.

    2. Mark a point with a pencil where you want the centre of your circle to be.

    3. Place the pointer of the compass there. Name it O.

    4. Turn the compass through a full arc swing to draw the circle. Make sure you complete the movement in one instant.

NCERT Problem 1: Draw a circle of radius 3.2 cm.

Solution: Follow the steps mentioned above. Take the radius as 3.2 cm and draw the circle. Mark the centre as O. You'll have your required figure.

A line segment has two endpoints, which means you can measure it using a ruler. Let us understand this with the help of a problem.

NCERT Problem 2: Construct a line segment of length 4.2 cm.

Method I: Using only a ruler.

    • Mark two points 4.2 cm from each other using a pencil and a ruler.
    • Join the two points to obtain the required line segment.

Method II: Using a compass.

    • Draw a line L of any length using a ruler—Mark a point A on the line.
    • Open the compass along the ruler such that the separation between the needle and pencil tip is 4.2 cm.
    • Press the needle at point A and draw an arc that cuts the line l.
    • Mark the point where it cuts the line as B.
    • AB is your required line segment.

To Construct a Copy of the Given Line Segment

Suppose you want to draw a line segment whose length is equal to that of a given line segment AB. The length of AB is not known.

There are many ways to do it.

Method I: Use a ruler.

    • Using a ruler, measure the length of line segment AB.
    • Draw a line segment of that length.

Method II: Using a transparent sheet.

    • Place a transparent sheet over AB.
    • Trace along AB to obtain your copy of the line segment.

Method III: Using a compass.

    • Fix the needle of your compass at A and your pencil tip at B. The compass opening is the length of AB.
    • Draw a line L of any length, using a ruler and pencil—Mark a point C on it.
    • Fix your needle at C, and draw an arc without disturbing the opening of the compass. Mark the point where it cuts the line L as D.
    • CD is your required line segment.

Two lines (or rays or segments) are perpendicular if they intersect at right angles. For example, each side of a square or rectangle intersects its adjacent sides perpendicularly.

Draw a perpendicular to a line through a point on it (using a compass and a scale)

    • Given a point P on a line L.
    • Taking P as a centre, construct an arc of any radius. Mark the points where it cuts the line L as A and B.
    • With A and B as centres, and radius greater than AP, construct two arcs to intersect. Mark their intersection as C.
    • Join P and C to get the desired perpendicular.

 

Perpendicular to a line through a point not on it (Using a compass and ruler)

    • Given a line L and a point P, not on it.
    • With P as a centre, draw an arc intersecting L at two points. Call these intersections A and B.
    • Without changing the radius, draw two arcs with A and B as centres on the other side of L. Call their intersection C.
    • Join P and C to obtain the desired perpendicular to L.

Perpendicular bisector of a line segment (using a compass and ruler)

    • Given a line segment AB of any length.
    • Taking A as a centre and a radius more than half the length of AB, draw a circle.
    • Using the same radius, draw a circle taking B as the centre. Suppose this new circle cuts the previous circle at points C and D.
    • Join C and D to obtain the desired perpendicular bisector.

Angles of Specific Measures

     1. Constructing a 60° angle

    • Draw a line L and mark a point O on it.
    • Place the compass needle at O. Taking any convenient radius, draw an arc. Mark the point where it cuts L as A.
    • Place the needle at A now. Using the same radius, draw an arc that cuts the previous arc. Mark the intersection as B.
    • Join O and B. OB will make an angle of 60° with L. 

     2. Constructing multiples of 60° angles.

    • Repeat the previous steps.
    • Now place the needle at B and, using the same radius, make another arc that cuts the original arc. Mark this point as D.
    • Join O and D. OD will make an angle of 120° with L.
    • Repeat for other multiples of 60°.

Angle Bisectors

    • Given an angle A.
    • With A as the centre, draw an arc that cuts both the rays of angle A. Mark the intersections as B and C.
    • Using the same radius, draw two arcs in the interior using B and C as centres. Mark the intersection of these arcs as D.
    • Join AD to get your desired angle bisector.

 

Construction of a copy of an angle of unknown measure

    • Given angle A of unknown measure.
    • Draw a line L and choose a point P on it.
    • Place the compass needle at A and draw an arc of convenient radius to cut the rays of angle A at B and C.
    • Use the same compass setting to draw an arc with P as a centre, cutting L in Q.
    • Set your compass to the length BC with the same radius.
    • Place the compass needle at Q and draw the arc to cut the arc drawn earlier in R.
    • Join P and R. It gives us angle P with the same measure as angle A.

 

Some Angles of Special Measures

NCERT Problem 3: How will you construct the following angles using a compass and a ruler?

30°

    • Construct an angle of 60° discussed above.
    • Bisect the angle using the method discussed above to obtain 30°.

 

45°

    • Construct an angle of 60°.
    • Construct an angle of 30°.
    • Bisect the angle between them. The line segment thus obtained will make an angle of 45° with the original line.

 

135°

    • Construct an angle of 180°.
    • Construct an angle of 120°.
    • Bisect the angle between them.

 

90°

    • Construct an angle of 60°
    • Construct an angle of 120°
    • Bisect the angle between them.

Being able to construct angles and shapes from essential geometrical tools is very important to understand the fundamentals of geometry. Most complicated mathematical figures are based on the simple shapes that students learn to construct in this chapter. Hence, practical geometry is an essential topic of mathematics. 

In-depth understanding is essential, whether in practical geometry or any other topic. That is what we at MSVgo strive to achieve. Our mission is to improve young students' mathematics skills and knowledge grasping capacity. We have a database of 15,000+ videos and 10,000+ questions, covering all the major educational boards in the country. We take pride in offering all of this at the cheapest price points. We make learning concise and fun for young learners.

So, why waste time? Head over to the MSVgo app right now, and leap towards a better future.

Other Courses

Related Chapters