The Triangle and Its Properties - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter explores the geometric properties of triangles including medians, altitudes, angle relationships, and an introduction to the Pythagoras theorem.
1. Why this chapter matters
The triangle is the simplest rigid polygon. Its properties are used in construction, engineering, navigation, and design. In CBSE exams, this chapter contributes 8-10 marks and is foundational for Class 8-10 geometry and trigonometry.
2. Medians of a triangle
A median connects a vertex to the midpoint of the opposite side.
- Every triangle has three medians.
- All three medians intersect at the centroid.
- The centroid divides each median in the ratio 2:1.
Properties of medians
A median is different from an altitude. A median always goes to the midpoint of the opposite side, while an altitude is perpendicular to the opposite side.
3. Altitudes of a triangle
An altitude is a perpendicular segment from a vertex to the line containing the opposite side.
- Every triangle has three altitudes.
- The point where altitudes meet is called the orthocenter.
- In a right triangle, two altitudes are the legs themselves.
Median vs altitude
| Property | Median | Altitude |
|---|---|---|
| Definition | Vertex to midpoint of opposite side | Vertex perpendicular to opposite side |
| Purpose | Bisects the opposite side | Measures height |
| Number | Always 3 | Always 3 |
| Point of concurrency | Centroid | Orthocenter |
4. Exterior angle property
An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
If triangle ABC has side BC extended to D, then: Angle ACD (exterior) = Angle A + Angle B
Application
This property is used to find unknown angles without measuring.
5. Angle sum property
The sum of the three interior angles of a triangle is always 180 degrees.
Angle A + Angle B + Angle C = 180 degrees.
This is true for EVERY triangle -- acute, obtuse, or right.
6. Types of triangles
Based on sides
- Equilateral triangle: All three sides equal. Each angle = 60.
- Isosceles triangle: Two sides equal. Angles opposite equal sides are equal.
- Scalene triangle: No sides equal.
Based on angles
- Acute triangle: All angles less than 90.
- Right triangle: One angle = 90.
- Obtuse triangle: One angle greater than 90.
7. Equilateral and isosceles triangles in detail
Equilateral triangle
- All sides equal.
- All angles equal to 60 degrees.
- The median, altitude, and angle bisector from any vertex are the same line.
Isosceles triangle
- Two sides equal (called legs).
- Angles opposite equal sides are equal (base angles).
- The altitude from the vertex between equal sides bisects the base.
8. Pythagoras theorem (introduction)
In a right triangle, the square of the hypotenuse (the longest side) equals the sum of squares of the other two sides.
If angle B = 90 in triangle ABC: AC x AC = AB x AB + BC x BC
Important notes
- The theorem applies ONLY to right triangles.
- The hypotenuse is the side opposite the right angle.
- A Pythagorean triplet satisfies a^2 + b^2 = c^2, e.g., (3, 4, 5) and (5, 12, 13).
Pythagorean triplets
| a | b | c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 6 | 8 | 10 | 36 + 64 = 100 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 8 | 15 | 17 | 64 + 225 = 289 |
9. Worked examples
Example 1: In triangle ABC, angle A = 65, angle B = 45. Find angle C.
Angle A + Angle B + Angle C = 180. 65 + 45 + C = 180. 110 + C = 180. C = 70.
Example 2: An exterior angle of a triangle is 120 and its interior opposite angles are equal. Find the angles.
Let each opposite interior angle = x. Exterior angle = sum of opposite interior angles = x + x = 2x = 120. x = 60. So interior opposite angles are 60 each. Third angle = 180 - 60 - 60 = 60. Therefore the triangle is equilateral.
Example 3: A right triangle has sides 9 cm and 12 cm. Find the hypotenuse.
Hypotenuse^2 = 9^2 + 12^2 = 81 + 144 = 225. Hypotenuse = sqrt(225) = 15 cm.
Example 4: In an isosceles triangle, the base angle is 40. Find the vertex angle.
Base angles are equal. Each base angle = 40. Sum of angles = 40 + 40 + vertex angle = 180. Vertex angle = 180 - 80 = 100.
10. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Confusing median and altitude | Median goes to midpoint; altitude is perpendicular |
| Using Pythagoras on non-right triangles | Pythagoras theorem works only for right triangles |
| Forgetting exterior angle = sum of OPPOSITE interior angles | The exterior angle uses the two NON-adjacent interior angles |
| Missing that triangle sum = 180 always | Every triangle, no exception, sums to 180 |
| Assuming all medians equal length | Medians can be different lengths in non-equilateral triangles |
11. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Angle sum property | 2-3 marks | 1 question |
| Exterior angle property | 2 marks | 1 question |
| Pythagoras theorem | 3 marks | 1 question |
| Median and altitude identification | 2 marks | 1 question |
| Equilateral/isosceles properties | 3 marks | Occasional |
12. Self-test
- Two angles of a triangle are 40 and 75. Find the third angle.
- In a right triangle, the two legs are 6 cm and 8 cm. Find the hypotenuse.
- The exterior angle of a triangle is 110 and one opposite interior angle is 60. Find the other interior angle.
- In an isosceles triangle, the vertex angle is 50. Find the base angles.
- State whether a triangle with sides 7 cm, 24 cm, and 25 cm is a right triangle.
- In triangle PQR, angle P = 2x, angle Q = 3x, angle R = 5x. Find all angles.
13. Answer key
- Third angle = 180 - (40 + 75) = 180 - 115 = 65.
- Hypotenuse^2 = 6^2 + 8^2 = 36 + 64 = 100. Hypotenuse = 10 cm.
- Exterior = sum of opposite interior. 110 = 60 + other. Other = 50.
- Let each base angle = x. 2x + 50 = 180. 2x = 130. x = 65. Base angles = 65 each.
- 7^2 + 24^2 = 49 + 576 = 625 = 25^2. Yes, it is a right triangle.
- 2x + 3x + 5x = 180. 10x = 180. x = 18. Angles: P = 36, Q = 54, R = 90.
14. Quick revision
- Median connects vertex to midpoint of opposite side.
- Altitude is perpendicular from vertex to opposite side.
- Sum of interior angles = 180 degrees.
- Exterior angle = sum of two opposite interior angles.
- Pythagoras theorem: hypotenuse^2 = sum of squares of legs (right triangles only).
- Equilateral: all sides/angles equal. Isosceles: two sides/angles equal.
