Practical geometry is the branch of mathematics in which we draw precise geometrical figures using instruments. While earlier chapters described shapes in words, this chapter teaches us to actually construct them accurately. The instruments we use are a ruler, a compass, a protractor and a set square. With these tools, we can draw lines, angles, circles and triangles with great precision.
The most important instrument is the compass, which is used to draw circles and arcs, and to copy or compare lengths. A compass has two arms: one with a sharp needle point and the other with a pencil. The distance between the needle and the pencil tip is the radius of the circle drawn. Before drawing, we must hold the compass correctly and keep the needle fixed at the centre.
In this chapter, we learn to draw circles of given radii, draw line segments of given lengths, copy line segments, draw perpendiculars to a line through a point on it or outside it, draw perpendicular bisectors of line segments, construct angles of 60, 120 and 90 degrees, and finally construct triangles given different sets of measurements. These skills build accuracy and confidence in geometry.
2. Instruments in the Geometry Box
The Ruler
A ruler is used to draw straight line segments and to measure lengths. It is marked in centimetres and millimetres.
The Compass
A compass is used to draw circles and arcs of a given radius. It can also be used to copy lengths and compare two lengths.
The Divider
A divider looks similar to a compass but has needles on both arms. It is used to compare and copy lengths accurately, and to step off equal divisions.
The Protractor
A protractor is a semicircular instrument marked from 0 to 180 degrees. It is used to measure and construct angles.
The Set Square
A set square is a triangular instrument used to draw perpendicular lines and parallel lines.
3. Drawing a Circle
A circle of a given radius is drawn using a compass.
Steps to Draw a Circle of Radius r
Open the compass to the required radius by measuring it on the ruler.
Mark a point O on the paper. This will be the centre of the circle.
Place the needle of the compass at O and press it gently to fix the centre.
Rotate the compass arm holding the pencil completely around to draw the circle.
Keep the radius fixed while rotating the compass.
The point O is the centre, and the distance from O to any point on the circle is the radius r.
4. Drawing a Line Segment
A line segment of a given length is drawn using a ruler.
Steps to Draw a Segment of Length 5.6 cm
Place the ruler on the paper and mark a point at the 0 cm mark. Name it A.
Mark another point at the 5.6 cm mark. Name it B.
Join points A and B with a straight line using the ruler edge.
The segment AB has length 5.6 cm.
Copying a Line Segment
To copy a segment, we can use the compass:
1. Measure the given segment AB with the compass.
2. Draw a ray or a new line from a point C.
3. Mark the measured length on the new line from C to get point D.
4. The new segment CD has the same length as AB.
5. Perpendiculars
A perpendicular is a line that meets another line at a right angle (90 degrees).
Perpendicular to a Line Through a Point on the Line
Place the protractor with its centre at the given point P on the line.
Mark 90 degrees with a point.
Join P to this point. This line is perpendicular to the given line at P.
Alternatively, use the compass:
1. With P as centre, draw arcs cutting the line at two points on both sides.
2. With each of these points as centre and the same radius, draw two arcs intersecting each other above or below.
3. Join the point of intersection to P. This is the perpendicular.
Perpendicular to a Line Through a Point Outside the Line
Similar steps are used. Draw arcs from the outside point to cut the line at two points, then draw intersecting arcs and join them.
6. Perpendicular Bisector of a Line Segment
The perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to it. Every point on the perpendicular bisector is at equal distance from the two endpoints.
Steps to Draw the Perpendicular Bisector of Segment AB
Open the compass to a radius more than half the length of AB.
With A as centre, draw arcs above and below the segment.
With B as centre and the same radius, draw arcs intersecting the previous arcs.
Join the two points of intersection. This line is the perpendicular bisector of AB.
The perpendicular bisector divides the segment into two equal parts and meets it at right angles.
7. Constructing Angles
Constructing a 60-degree Angle
Draw a ray OA.
With O as centre and any convenient radius, draw an arc cutting OA at point P.
With P as centre and the same radius, draw another arc cutting the first arc at point Q.
Join O and Q. The angle AOQ is 60 degrees.
Constructing a 120-degree Angle
Using the same construction, the other angle formed is 120 degrees. Alternatively, extend the steps: draw a third arc with the same radius to get an angle of 120 degrees.
Constructing a 90-degree Angle
Construct a perpendicular to a line at a point, which gives an angle of 90 degrees. Alternatively, using a compass, construct 60 degrees and then bisect the supplementary angle, or draw the perpendicular bisector.
Bisecting an Angle
To bisect an angle, place the compass at the vertex, draw an arc cutting both arms, then draw two arcs from the two cut points that intersect each other. Join the vertex to this intersection. This ray divides the angle into two equal halves.
8. Constructing Triangles
A triangle can be constructed when we know certain measurements. The three cases are:
Case 1: Three sides given (SSS)
Draw one side, say BC, of the given length.
With B as centre and the length of AB as radius, draw an arc.
With C as centre and the length of AC as radius, draw another arc intersecting the first arc at A.
Join A to B and A to C. Triangle ABC is ready.
Case 2: Two sides and the included angle given (SAS)
Draw one side AB.
At A, construct the given angle using a protractor.
On the other arm of the angle, mark the second given length AC.
Join C to B. Triangle ABC is ready.
Case 3: Two angles and the included side given (ASA)
Draw the given side AB.
At A, construct the first given angle.
At B, construct the second given angle.
Extend the two rays until they meet at C. Triangle ABC is ready.
Case 4: Right-angled triangle with hypotenuse and one side
Draw the given leg, say BC.
At C, construct a right angle.
With B as centre and the hypotenuse as radius, cut the perpendicular arm at A.
Join A and B. The right-angled triangle ABC is ready.
Quick Revision Tables
Instrument
Use
Ruler
Draw and measure line segments
Compass
Draw circles and arcs, copy lengths
Divider
Compare and transfer lengths
Protractor
Measure and draw angles
Set square
Draw perpendiculars and parallels
Construction
Method
60-degree angle
Arc with same radius twice
90-degree angle
Perpendicular at a point
Perpendicular bisector
Arcs from both ends, join intersections
Triangle with 3 sides
Arcs from two ends, join to intersection
Mind Map
flowchart TD
A["Practical Geometry"] --> B["Instruments"]
A --> C["Drawing a Circle"]
A --> D["Drawing Line Segments"]
A --> E["Perpendiculars"]
A --> F["Perpendicular Bisector"]
A --> G["Constructing Angles"]
A --> H["Constructing Triangles"]
B --> B1["Ruler, compass, divider, protractor, set square"]
E --> E1["Through a point on the line"]
E --> E2["Through a point outside the line"]
G --> G1["60, 90, 120 degrees"]
G --> G2["Angle bisector"]
H --> H1["SSS, SAS, ASA"]
H --> H2["Right-angled triangle"]
F --> F1["Midpoint + right angle"]
Important Diagrams (SVG)
Circle with Centre and Radius
Perpendicular Bisector of Segment AB
Common Mistakes
Keeping the compass needle moving while drawing a circle, which changes the radius.
Measuring the radius from the wrong mark on the ruler.
Forgetting to keep the radius more than half the segment length when drawing the perpendicular bisector.
Drawing arcs with different radii from the two endpoints when constructing the perpendicular bisector.
Reading the wrong scale on the protractor when constructing angles.
In triangle construction, joining the wrong points or using the wrong given measurements.
Exam Tips
Practise constructions with a compass and ruler before the exam so your drawings are clean and accurate.
Always mark the centre, endpoints and key points clearly with letters.
In every construction, show the construction arcs lightly so the examiner can see the method.
Remember the compass radius must be more than half the segment for the perpendicular bisector.
For angle construction, remember the 60-degree arc construction: same radius used twice.
Write the construction steps briefly in your own words for full marks.
Conclusion
Practical Geometry turns abstract shapes into accurately drawn figures using a ruler, compass, protractor, divider and set square. We learnt to draw circles and line segments, copy lengths, construct perpendiculars and perpendicular bisectors, draw standard angles of 60, 90 and 120 degrees, and construct triangles using SSS, SAS and ASA conditions. These hands-on skills develop precision, logical thinking and the ability to visualise geometry, which will be invaluable in higher classes.