Comprehensive theory, key formulas, diagrams, and memory aids for Practical Geometry.
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.
A ruler is used to draw straight line segments and to measure lengths. It is marked in centimetres and millimetres.
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.
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.
A protractor is a semicircular instrument marked from 0 to 180 degrees. It is used to measure and construct angles.
A set square is a triangular instrument used to draw perpendicular lines and parallel lines.
A circle of a given radius is drawn using a compass.
The point O is the centre, and the distance from O to any point on the circle is the radius r.
A line segment of a given length is drawn using a ruler.
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.
A perpendicular is a line that meets another line at a right angle (90 degrees).
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.
Similar steps are used. Draw arcs from the outside point to cut the line at two points, then draw intersecting arcs and join them.
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.
The perpendicular bisector divides the segment into two equal parts and meets it at right angles.
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.
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.
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.
A triangle can be constructed when we know certain measurements. The three cases are:
| 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 |
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"]
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.