1. Chapter at a Glance
How long a ladder do you need to reach a window 4 m high if its foot is 3 m from the wall? How long is the diagonal of a TV screen? One of the most famous results in all of mathematics answers such questions: in a right-angled triangle, the square on the longest side equals the sum of the squares on the other two sides. It was stated in India by Baudhāyana in the Śulba-sūtras (written for building fire altars, roughly 800–500 BCE) and is also linked with the Greek thinker Pythagoras. NCERT calls it the Baudhāyana–Pythagoras theorem.
- Hypotenuse and legs of a right triangle.
- The statement of the theorem and Baudhāyana's original form.
- Proofs by rearranging areas.
- The converse — testing whether a triangle is right-angled.
- Pythagorean triples.
- Applications: ladders, diagonals, distances.
2. The Theorem
In a right triangle, the side opposite the right angle is the hypotenuse (the longest side); the other two sides are the legs.
If a triangle has legs a and b and hypotenuse c, then a² + b² = c².
Baudhāyana's statement (in words): the diagonal rope of a rectangle produces as much area as the two sides produce separately. In other words, the square on the diagonal of a rectangle equals the squares on its length and breadth together.
3. A Proof by Rearranging Areas
Take a big square of side (a + b). Place four copies of the right triangle (legs a, b, hypotenuse c) inside it, one in each corner, so that their hypotenuses form a tilted square in the middle.
- Area of big square = (a + b)² = a² + 2ab + b².
- It is also = 4 triangles + inner square = 4 × ½ab + c² = 2ab + c².
- So a² + 2ab + b² = 2ab + c², which gives a² + b² = c².
4. The Converse
If the sides of a triangle satisfy a² + b² = c² (c the longest side), then the triangle is right-angled, with the right angle opposite c.
Builders in ancient India and Egypt used this: a rope with knots dividing it into 3 + 4 + 5 = 12 equal parts, stretched into a triangle, gives a perfect right angle.
Is a triangle with sides 7 cm, 24 cm, 25 cm right-angled?
Solution: 7² + 24² = 49 + 576 = 625 = 25². Yes, the right angle is opposite the 25 cm side.
If a² + b² > c², the largest angle is acute; if a² + b² < c², it is obtuse.
5. Pythagorean Triples
Whole numbers a, b, c with a² + b² = c² form a Pythagorean triple.
| Triple | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 8, 15, 17 | 64 + 225 = 289 |
| 7, 24, 25 | 49 + 576 = 625 |
| 20, 21, 29 | 400 + 441 = 841 |
| 6, 8, 10 (2 × 3,4,5) | 36 + 64 = 100 |
- Multiplying a triple by any number gives another triple.
- For any m > 1: (2m, m² − 1, m² + 1) is a triple. m = 4 gives (8, 15, 17).
6. Applications
A 10 m ladder reaches a window 8 m high. How far is its foot from the wall?
Solution: d² + 8² = 10² → d² = 36 → d = 6 m.
Find the diagonal of a rectangle 15 cm × 8 cm.
Solution: √(225 + 64) = √289 = 17 cm.
Ravi walks 12 km north and then 5 km east. How far is he from the start?
Solution: √(144 + 25) = √169 = 13 km.
Note: when the sides are not perfect squares, the answer may be a surd, e.g. the diagonal of a 1 × 1 square is √2 ≈ 1.414. Baudhāyana gave a remarkably accurate value of √2 in the Śulba-sūtras.
7. Solved Problems (NCERT Style)
The hypotenuse of a right triangle is 26 cm and one leg is 10 cm. Find the other leg.
Solution: √(676 − 100) = √576 = 24 cm.
A tree breaks at a height of 5 m and its top touches the ground 12 m from the base. Find the original height.
Solution: Broken part = √(25 + 144) = 13 m. Height = 5 + 13 = 18 m.
Each leg of an isosceles right triangle is 7 cm. Find the hypotenuse.
Solution: √(49 + 49) = √98 = 7√2 ≈ 9.9 cm.
8. Master Formula Table
| Result | Formula |
|---|---|
| Theorem | a² + b² = c² |
| Hypotenuse | c = √(a² + b²) |
| A leg | a = √(c² − b²) |
| Rectangle diagonal | √(l² + b²) |
| Square diagonal | side × √2 |
| Triple generator | (2m, m² − 1, m² + 1) |
| Angle test | a² + b² vs c²: =, >, < → right, acute, obtuse |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Theorem | Square on hypotenuse = sum of squares on legs | Find a side |
| History | Baudhāyana's Śulba-sūtra | Indian contribution |
| Proof | Area rearrangement | Explain |
| Converse | Test for a right angle | Check given sides |
| Triples | Whole-number solutions | Generate, recognise |
| Applications | Ladders, diagonals, walks | Word problems |
10. Mind Map
- a² + b² = c²
- Hypotenuse longest
- Area proof
- Śulba-sūtra
- Right-angle test
- Knotted rope 3-4-5
- Triples
- Ladders, diagonals
11. Tips and Tricks
- Remember common triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples — many questions use them.
- Always identify the hypotenuse first: it is opposite the right angle and is the longest side.
- Draw a sketch for word problems.
- Adding the squares when finding a leg (you must subtract from c²).
- Using the theorem in a triangle that is not right-angled.
- Forgetting to take the square root at the end.
12. Important One-liners
- In a right triangle, hypotenuse² = leg² + leg².
- The hypotenuse is the longest side.
- Baudhāyana stated the result in the Śulba-sūtras for altar construction.
- If a² + b² = c², the triangle is right-angled.
- 3, 4, 5 is the smallest Pythagorean triple.
- The diagonal of a square of side s is s√2.
13. Exam-pattern Questions with Detailed Answers
The legs of a right triangle are 6 cm and 8 cm. The hypotenuse is:
- 7 cm
- 10 cm
- 14 cm
- 12 cm
Show answer
√(36 + 64) = √100 = 10.
Answer: (B) 10 cm
Is (9, 12, 15) a Pythagorean triple?
Show answer
81 + 144 = 225 = 15². Yes (it is 3 × (3, 4, 5)).
Answer: Yes
Find the diagonal of a square of side 5 cm (to 1 decimal place).
Show answer
5√2 ≈ 5 × 1.414 = 7.07 ≈ 7.1 cm.
Answer: ≈ 7.1 cm
A 13 m ladder leans against a wall with its foot 5 m from the wall. How high does it reach?
Show answer
√(169 − 25) = √144 = 12 m.
Answer: 12 m
Which set forms a right triangle?
- 4, 5, 6
- 5, 12, 13
- 6, 7, 9
- 2, 3, 4
Show answer
25 + 144 = 169 = 13².
Answer: (B)
Write a Pythagorean triple with 18 as the smallest even member using (2m, m² − 1, m² + 1).
Show answer
2m = 18 → m = 9: (18, 80, 82).
Answer: 18, 80, 82
A TV screen is 40 inches wide and 30 inches tall. Find its diagonal.
Show answer
√(1600 + 900) = √2500 = 50 inches.
Answer: 50 inches
A triangle has sides 9, 10 and 14. Is its largest angle acute, right or obtuse?
Show answer
81 + 100 = 181 < 196 = 14². So the angle opposite 14 is obtuse.
Answer: Obtuse
Two poles 6 m and 11 m tall stand 12 m apart. Find the distance between their tops.
Show answer
Height difference 5 m, horizontal gap 12 m. Distance = √(25 + 144) = 13 m.
Answer: 13 m
The diagonals of a rhombus are 30 cm and 16 cm. Find its perimeter.
Show answer
Half-diagonals 15 and 8 (they meet at 90°). Side = √(225 + 64) = 17. Perimeter = 68 cm.
Answer: 68 cm
A rope is divided by knots into 12 equal parts. Show how it can be used to make a right angle.
Show answer
Form a triangle with sides 3, 4 and 5 parts. Since 3² + 4² = 5², the converse shows the angle between the 3 and 4 sides is 90°.
Answer: 3 + 4 + 5 = 12 parts; 3-4-5 triangle is right-angled
A plane flies 320 km east and then 240 km north. How far is it from its starting point?
Show answer
320 = 80 × 4, 240 = 80 × 3 → 3-4-5 triple scaled by 80 → 400 km.
Answer: 400 km
14. Quick Revision Summary
- Right triangle: a² + b² = c² (c = hypotenuse, opposite 90°).
- Stated by Baudhāyana in the Śulba-sūtras; proved by area rearrangement.
- Converse tests for right angles; compare a² + b² with c² for acute/obtuse.
- Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25; generator (2m, m² − 1, m² + 1).
- Use for ladders, diagonals, distances and heights.