Triangles – Class 10 CBSE Mathematics
(Study notes in plain, everyday language)
1. What is a Triangle?
- A triangle is a shape made of three straight sides joined end‑to‑end.
- Think of a pizza slice: three edges make a triangular piece.
Tiny Example
Draw a line AB, a line BC, and a line CA.
These three lines form a triangle ABC.
2. Naming a Triangle
- The letters that name the corners (vertices) are written in clockwise or counter‑clockwise order.
- Example: Triangle ABC means side AB, side BC, side CA.
3. Types of Triangles
| How we look at it | Everyday Analogy | Triangle’s Name |
|---|---|---|
| Side lengths | Three different sticks – a “scalene” triangle. | Scalene |
| Two sticks of the same length – a “isosceles” triangle. | Isosceles | |
| All sticks equal – a “equilateral” triangle. | Equilateral | |
| Angles | One angle turns 90° – a “right” triangle. | Right |
| All angles are 60° – an “acute” triangle. | Acute | |
| One angle is >90° – a “obtuse” triangle. | Obtuse |
4. Basic Properties
| Property | Simple Explanation | Quick Check |
|---|---|---|
| Sum of angles | All angles inside a triangle always add up to 180°. | Add the three angles you see. |
| Largest side opposite largest angle | The longer a side, the larger the angle it faces. | Compare a side with the angle opposite it. |
| Shortest side opposite shortest angle | The shorter a side, the smaller the angle it faces. | Compare a side with the angle opposite it. |
5. Pythagoras’ Theorem (Right Triangles Only)
- In a right triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides.
- Everyday: If you build a square on each side, the big square on the hypotenuse equals the two smaller squares together.
Formula
c² = a² + b²
- c = hypotenuse (longest side)
- a, b = the other two sides
Tiny Example
Right triangle with sides 3 cm, 4 cm, and 5 cm.
Check: 5² = 25, 3² + 4² = 9 + 16 = 25 → ✔️
6. Area of a Triangle
| When you know | Formula | Quick Tip |
|---|---|---|
| Base and height | ½ × base × height | Think of the triangle as half of a rectangle. |
| All three sides (Heron’s formula) | ½ × √[s(s‑a)(s‑b)(s‑c)] | s is half the perimeter. (Only for later classes.) |
Tiny Example (base = 6 cm, height = 4 cm)
Area = ½ × 6 × 4 = 12 cm²
7. Finding Missing Sides or Angles
7.1 Using the Angle Sum
If you know two angles, the third = 180° – (sum of the two).
Tiny Example
Angles 50° and 60° → third = 180 – 110 = 70°
7.2 Using Pythagoras (Right Triangle)
If two sides are known → find third as shown above.
7.3 Using Proportions (Similar Triangles)
If two triangles look the same (same shape, just different size), corresponding sides are in the same ratio.
Tiny Example
In two similar triangles, one side is 4 cm and the corresponding side is 6 cm → the ratio is 4:6 = 2:3.
If another side in the first triangle is 8 cm, the matching side in the second is 8 × 3/2 = 12 cm.
8. Quick Practice Problems
| Problem | What to Find | Hint |
|---|---|---|
| 1. In triangle ABC, AB = 5 cm, BC = 12 cm, and ∠ABC = 90°. Find AC. | Hypotenuse | Use Pythagoras. |
| 2. A triangle has angles 40°, 60°, and ?. | Third angle | Angle Sum. |
| 3. Two similar triangles: small one has side 3 cm, large one has the same side 9 cm. What is a side that is 5 cm in the small triangle? | Corresponding side in large triangle | Scale factor = 9/3 = 3. |
| 4. Triangle has base 8 cm and height 5 cm. Find its area. | Area | ½ × base × height. |
9. Key Take‑Away Statements
- A triangle = 3 straight sides.
- 3 angles always add to 180°.
- Right triangle: longest side² = sum of other two sides².
- Area = ½ × base × height.
- In similar triangles, sides are in the same proportion.
10. Visual Check (Sketch the following)
C
/\
/ \
/ \
/______\
A B
- Label sides AB, BC, CA.
- Label angles ∠A, ∠B, ∠C.
- If ∠B = 90°, it is a right triangle.
Good luck! Practice drawing triangles, labeling sides and angles, and solving the quick problems above. The more you draw, the easier the concepts become.