📚 Class 7 CBSE – Mathematics
The Triangle & Its Properties
A triangle is a 3‑sided polygon. Its properties form the foundation for many later topics in geometry and trigonometry.
1. Simple Explanation
| Concept | What it means |
|---|---|
| Vertices | The three corner points (A, B, C). |
| Sides | The three line segments (AB, BC, CA). |
| Angles | Three interior angles (∠A, ∠B, ∠C). |
| Exterior angle | Angle formed by extending one side; equals the sum of the two opposite interior angles. |
| Perimeter | Total length of the three sides. |
| Area | Space inside the triangle. |
Key idea:
The angles of any triangle always add up to 180° (or π radians).
This fact is the cornerstone for classifying triangles and proving many of their properties.
2. Key Points
2.1 Classification by Sides
| Type | Definition | Example |
|---|---|---|
| Equilateral | All sides equal (a = b = c). | All angles 60°. |
| Isosceles | Two sides equal (a = b ≠ c). | Two base angles equal. |
| Scalene | All sides different (a ≠ b ≠ c). | All angles different. |
2.2 Classification by Angles
| Type | Angle condition | Example |
|---|---|---|
| Acute | All angles < 90°. | 70°, 60°, 50°. |
| Right | One angle = 90°. | 90°, 60°, 30°. |
| Obtuse | One angle > 90°. | 120°, 30°, 30°. |
2.3 Triangle Inequality Theorem
For any triangle with sides a, b, c:
- a + b > c
- b + c > a
- c + a > b
If any of these fail, the “three sides” cannot form a triangle.
2.4 Area Formulas
| Situation | Formula | Example |
|---|---|---|
| Base × Height | (\displaystyle \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}) | ( \frac12 \times 6 \text{ cm} \times 4 \text{ cm} = 12 \text{ cm}^2) |
| Heron’s Formula (all sides known) | (\displaystyle \text{Area} = \sqrt{s(s-a)(s-b)(s-c)}) where (s=\frac{a+b+c}{2}) | For 5‑12‑13 triangle: (s=15), area = 30 |
| Right Triangle | (\displaystyle \text{Area} = \frac12 \times \text{leg}_1 \times \text{leg}_2) | 5‑12‑13: area = 30 |
2.5 Important Theorems
| Theorem | Statement |
|---|---|
| Pythagoras | In a right triangle, (a^2 + b^2 = c^2) (hypotenuse² = sum of squares of legs). |
| Exterior Angle Theorem | An exterior angle equals the sum of the two opposite interior angles. |
| Angle Bisector Theorem | The angle bisector divides the opposite side proportionally to the adjacent sides. |
| Median to the Hypotenuse | In a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices. |
2.6 Common Terminology
- Altitude – perpendicular from a vertex to the opposite side (or its extension).
- Median – segment from a vertex to the midpoint of the opposite side.
- Angle Bisector – segment that divides an angle into two equal parts.
- Circumcenter – intersection of perpendicular bisectors of sides.
- Incenter – intersection of angle bisectors.
(These will be covered in more detail in later classes.)
3. Worked Example
Problem
Find the type, perimeter, and area of a triangle whose sides are 5 cm, 12 cm, and 13 cm.
Step‑by‑Step
-
Check Triangle Inequality
- 5 + 12 = 17 > 13 ✔️
- 12 + 13 = 25 > 5 ✔️
- 13 + 5 = 18 > 12 ✔️
→ A triangle can be formed.
-
Classify by Sides
- All sides different → Scalene.
-
Classify by Angles (Pythagoras)
- Check if (5^2 + 12^2 = 13^2).
- (25 + 144 = 169) → equality holds.
→ One angle is