Practical Geometry - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter focuses on hands-on geometric constructions using ruler and compass, building precision and spatial reasoning skills.
1. Why this chapter matters
Practical geometry develops the ability to construct accurate figures using basic tools. These skills are applied in engineering, architecture, and design. In CBSE exams, this chapter contributes 6-8 marks with stepwise construction questions that test both accuracy and understanding.
2. Tools required
- Ruler (unmarked for straight lines, marked for measurements)
- Compass (for arcs and circles)
- Protractor (for measuring angles)
- Pencil (sharp for precision)
3. Construction of parallel lines
Method 1: Using ruler and set square
To draw a line parallel to a given line AB through a point P:
- Place one side of set square along AB.
- Place ruler against the other side of set square.
- Hold ruler fixed. Slide set square along ruler until it touches P.
- Draw line through P.
Method 2: Using compass (angle copy method)
To draw a line parallel to AB through point P:
- Draw a transversal through P intersecting AB at any angle.
- At P, copy the angle formed at the intersection point.
- The line through P making the copied angle is parallel to AB.
4. Construction of triangles
General conditions for triangle construction
A triangle can be constructed if:
- Three sides are given (SSS).
- Two sides and the included angle are given (SAS).
- Two angles and the included side are given (ASA).
- The hypotenuse and one side of a right triangle are given (RHS).
A triangle CANNOT be constructed if:
- Three angles are given (AAA) -- infinite triangles possible.
- Two sides and a non-included angle are given (SSA) -- ambiguous case.
5. Constructing a triangle given three sides (SSS)
Steps to construct triangle ABC with AB = 5 cm, BC = 6 cm, CA = 7 cm:
- Draw base BC = 6 cm.
- With B as centre, radius 5 cm, draw an arc.
- With C as centre, radius 7 cm, draw another arc intersecting the first arc at A.
- Join AB and AC.
Triangle inequality check
The sum of any two sides must be greater than the third side. 5 + 6 > 7, 5 + 7 > 6, 6 + 7 > 5. Construction is possible.
6. Constructing a triangle given two sides and included angle (SAS)
Steps to construct triangle ABC with AB = 4 cm, AC = 5 cm, angle A = 60:
- Draw AB = 4 cm.
- At A, construct angle 60 using a protractor.
- On the ray, mark point C such that AC = 5 cm.
- Join BC.
7. Constructing a triangle given two angles and included side (ASA)
Steps to construct triangle ABC with BC = 5 cm, angle B = 50, angle C = 60:
- Draw base BC = 5 cm.
- At B, construct angle 50.
- At C, construct angle 60. The rays intersect at A.
- Triangle ABC is complete.
Checking angle sum
Angle A = 180 - (50 + 60) = 70. The construction must be accurate.
8. Constructing a right triangle (RHS)
Steps to construct right triangle ABC with hypotenuse AC = 6 cm and side AB = 4 cm, right angle at B:
- Draw AB = 4 cm.
- At B, construct a perpendicular (90 degrees).
- With A as centre and radius 6 cm, draw an arc intersecting the perpendicular at C.
- Join AC.
9. Criteria summary table
| Criterion | Given elements | Steps summary | Validity check |
|---|---|---|---|
| SSS | 3 sides | Draw base, arcs from ends | Sum of any two sides > third |
| SAS | 2 sides + included angle | Draw base, construct angle, mark side | Angle must be between the two sides |
| ASA | 2 angles + included side | Draw base, construct both angles | Sum of given angles < 180 |
| RHS | Hypotenuse + 1 side | Draw side, right angle, arc from other end | Hypotenuse > given side |
10. Worked examples
Example 1: Construct triangle PQR with PQ = 4 cm, QR = 5 cm, RP = 6 cm. Write steps.
- Draw QR = 5 cm.
- With Q centre, radius 4 cm, draw arc above QR.
- With R centre, radius 6 cm, draw arc intersecting at P.
- Join PQ and PR. Check: 4 + 5 > 6, 4 + 6 > 5, 5 + 6 > 4. Valid.
Example 2: Construct triangle XYZ with XY = 5 cm, YZ = 4 cm, angle Y = 90.
- Draw XY = 5 cm.
- At Y, construct 90 angle using protractor or compass.
- On the ray, mark Z such that YZ = 4 cm.
- Join XZ. This is actually an SAS construction.
Example 3: Construct triangle ABC where angle A = 45, angle B = 75, AB = 6 cm.
- Draw AB = 6 cm.
- At A, construct angle 45.
- At B, construct angle 75.
- Rays intersect at C. Angle C = 180 - (45 + 75) = 60. Construction is valid.
11. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Arcs not intersecting due to incorrect radius | Check triangle inequality before starting |
| Angle constructed at wrong vertex | Label vertices clearly and construct at correct point |
| Using non-included angle in SAS | SAS needs the angle BETWEEN the two given sides |
| Ruler slipping during construction | Hold ruler firmly or use a set square as guide |
| Not showing construction arcs in exam | Always leave arcs visible for partial marks in CBSE |
12. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| SSS triangle construction | 3 marks | 1 question |
| SAS triangle construction | 3 marks | 1 question |
| ASA triangle construction | 3 marks | 1 question |
| RHS triangle construction | 3 marks | 1 question |
| Parallel line construction | 2 marks | 1 question |
13. Self-test
- Construct a triangle with sides 4 cm, 5 cm, 6 cm. Write the steps.
- Construct triangle ABC with AB = 5 cm, AC = 4 cm, angle A = 70.
- Construct triangle PQR with PQ = 6 cm, angle P = 55, angle Q = 65.
- Construct a right triangle with hypotenuse 7 cm and one side 4 cm.
- Can a triangle with sides 2 cm, 3 cm, 6 cm be constructed? Why?
- Draw a line AB = 8 cm. Mark point P 3 cm above it. Draw a line through P parallel to AB using compass method.
14. Answer key
1-4: Follow construction steps as described in sections 5-8. Accuracy of measurement and arc intersection is key. 5. No. 2 + 3 = 5, which is not greater than 6. Triangle inequality fails. 6. Follow the angle copy method described in section 3 (Method 2).
15. Quick revision
- Parallel lines: use set square sliding or angle copying.
- Triangle can be constructed if enough independent measurements are given.
- SSS: arcs from both ends of base.
- SAS: angle between given sides.
- ASA: given side between the two angles.
- RHS: for right triangles only.
- Always check triangle inequality for SSS.
- Leave construction arcs visible in exam answers.
