Practical Geometry
1. Basic Construction Tools
| Tool | Use |
|---|---|
| Ruler | Drawing straight lines, measuring lengths |
| Compass | Drawing arcs and circles, copying lengths |
| Protractor | Measuring and drawing angles |
| Set squares | Drawing perpendicular and parallel lines |
| Divider | Transferring lengths |
Precautions
- Keep pencil SHARP (thin lines).
- Use a RULER, not the edge of the notebook.
- Do NOT erase construction lines after the construction.
- Mark ALL given measurements clearly.
2. Construction of Parallel Lines
Method 1: Using Set Squares
- Place one set square along the given line.
- Place the second set square against the first.
- Hold the second fixed. Slide the first along.
- Draw the line along the edge of the first set square.
Method 2: Using Compass and Ruler
'Draw a line parallel to AB through point P outside the line.'
Steps:
- Take any point Q on AB.
- Join PQ.
- At point P, construct ∠QPY = ∠PQB (alternate interior angles equal).
- Extend PY. This is the required line parallel to AB.
Worked Example (ICSE 2024, 2 marks)
'Draw a line AB = 6 cm. Take a point P at 3 cm above AB. Draw a line through P parallel to AB.'
Solution: Follow method 1 or 2 above.
3. Construction of Triangles
Case 1: SSS Triangle (Given All 3 Sides)
'Construct ΔABC with AB = 5 cm, BC = 4 cm, AC = 6 cm.'
Steps:
- Draw AB = 5 cm.
- With centre A and radius 6 cm, draw an arc.
- With centre B and radius 4 cm, draw an arc cutting the first arc at C.
- Join AC and BC. ΔABC is constructed.
Case 2: SAS Triangle (Given 2 Sides and Included Angle)
'Construct ΔABC with AB = 5 cm, AC = 4 cm, and ∠A = 60°.'
Steps:
- Draw AB = 5 cm.
- At A, construct ∠BAX = 60° using protractor.
- On AX, cut AC = 4 cm.
- Join BC. ΔABC is constructed.
Case 3: ASA Triangle (Given 2 Angles and Included Side)
'Construct ΔABC with AB = 5 cm, ∠A = 50°, ∠B = 60°.'
Steps:
- Draw AB = 5 cm.
- At A, construct ∠BAX = 50°.
- At B, construct ∠ABY = 60°.
- AX and BY intersect at C. ΔABC is constructed.
4. Construction of Perpendicular Bisector
The perpendicular bisector of a segment is the line that is PERPENDICULAR to the segment and passes through its MIDPOINT.
'Draw the perpendicular bisector of AB = 6 cm.'
Steps:
- Draw AB = 6 cm.
- With centre A and radius > 3 cm (say 4 cm), draw arcs above and below AB.
- With the SAME radius and centre B, draw arcs cutting the previous arcs at P and Q.
- Join PQ. PQ is the perpendicular bisector of AB. It meets AB at its midpoint.
Properties
- Every point on the perpendicular bisector is EQUIDISTANT from A and B.
- It divides the line into TWO EQUAL parts.
5. Construction of Angle Bisector
The angle bisector divides an angle into TWO EQUAL parts.
'Construct the bisector of ∠ABC = 60°.'
Steps:
- Draw ∠ABC = 60°.
- With centre B and ANY radius, draw an arc cutting BA at P and BC at Q.
- With centre P and radius > half of PQ, draw an arc inside the angle.
- With the SAME radius and centre Q, draw an arc cutting the previous arc at R.
- Join BR. BR is the angle bisector. ∠ABR = ∠RBC = 30°.
6. Construction of 60°, 120°, 90° Angles Using Compass
60° Angle
- Draw a ray AB.
- With centre A and any radius, draw an arc cutting AB at P.
- With centre P and the SAME radius, draw an arc cutting the first arc at Q.
- Join AQ. ∠QAB = 60°.
120° Angle
Construct 60° twice, or construct 2 consecutive arcs of 60° each.
90° Angle
- Construct perpendicular bisector of a segment, OR
- Construct 60° and then bisect the remaining 60° to get 30° (60° + 30° + = 90° — actually construct 60°, then mark an arc of 60° again to get 120°, bisect the 60° between 90° and the original 60° — simpler: construct a perpendicular).
7. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| SSS triangle construction | 3 marks | Very High |
| SAS triangle construction | 3 marks | High |
| ASA triangle construction | 3 marks | High |
| Perpendicular bisector | 2-3 marks | High |
| Angle bisector | 2 marks | Medium |
| Parallel lines construction | 2-3 marks | Medium |
Common Mistakes
- Using wrong radius for arcs (especially for perpendicular bisector — radius must be > half the segment).
- Not labelling points clearly.
- Erasing construction arcs (DO NOT erase — they are part of the answer).
- In ASA, confusing included side with non-included side.
Self-Test (5 Questions)
Q1. What is the minimum radius for arcs when constructing a perpendicular bisector of a 5 cm segment? (1 mark)
Q2. 'Construct ΔABC with AB = 4.5 cm, BC = 5 cm, AC = 6 cm.' (3 marks)
Q3. 'Construct an angle of 60° using a compass.' (2 marks)
Q4. 'Draw a line segment PQ = 7 cm. Construct its perpendicular bisector.' (2 marks)
Q5. 'Construct ΔABC with AB = 5 cm, ∠A = 45°, ∠B = 75° using ASA criterion.' (3 marks)
Answers
A1. Radius > 2.5 cm (more than half of 5 cm). A2. Steps: Draw AB = 4.5 cm. Arc from A radius 6 cm. Arc from B radius 5 cm. Intersection = C. Join AC and BC. A3. Steps: Draw a ray. Arc from vertex. Same radius arc from intersection point on ray. Join vertex to second arc intersection. A4. Steps as described in Section 4. A5. Draw AB = 5 cm. At A, draw 45°. At B, draw 75°. They intersect at C.
