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Chapter 9 · NCERT 2025-26

📐 The Baudhayana–Pythagoras Theorem

Ancient Indian Roots · a² + b² = c² · Pythagorean Triples · Real-World Applications

a b c a² + b² = c²
📜 Historical Context & Indian Heritage

The relationship between the sides of a right-angled triangle is one of the most important discoveries in all of mathematics. While this theorem is widely known as the "Pythagorean Theorem" after the Greek mathematician Pythagoras (c. 570–495 BCE), the truth is that this knowledge was recorded in India centuries earlier.

🇮🇳 Baudhayana & the Sulba Sutras

Around 800 BCE, the Indian mathematician and Vedic priest Baudhayana composed the Baudhayana Sulba Sutra, one of the oldest mathematical texts in the world. The Sulba Sutras (literally "rope rules") were manuals for constructing fire altars of precise geometric shapes required for Vedic rituals.

In the Baudhayana Sulba Sutra (verse 1.48), we find this remarkable statement:

"The diagonal of a rectangle produces both areas which its length and breadth produce separately." Baudhayana Sulba Sutra, ~800 BCE — the earliest known statement of the theorem

In modern language, this means: The square on the diagonal of a rectangle equals the sum of the squares on its two sides. This is exactly the theorem we study today!

📜 Baudhayana (~800 BCE)

Composed the Sulba Sutras containing the first recorded statement of the theorem about right triangles. He also gave an approximation of √2 as 1 + 1/3 + 1/(3×4) − 1/(3×4×34) = 1.4142156..., which is accurate to 5 decimal places!

📜 Apastamba (~600 BCE)

Another Indian mathematician who independently stated the theorem and provided numerical examples of Pythagorean triples, including (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37).

🇨🇳 Ancient China (~1100 BCE)

The Chinese text Zhoubi Suanjing contains a similar statement and a visual proof of the 3-4-5 triangle, showing that knowledge of this theorem was widespread in the ancient world.

🇬🇷 Pythagoras (~570 BCE)

The Greek mathematician Pythagoras and his school are credited with providing a formal proof of the theorem around 500 BCE. The theorem is named after him in Western tradition, though the knowledge predates him.
💡 Why "Baudhayana–Pythagoras"? The NCERT Ganita textbook (2025-26) uses the name Baudhayana–Pythagoras Theorem to honour the Indian mathematician Baudhayana who described this relationship approximately 300 years before Pythagoras. This acknowledges India's significant contribution to world mathematics.
📚 The Sulba Sutras — Mathematical Treasure

The Sulba Sutras are remarkable not just for the theorem but for several other mathematical achievements:

  • Pythagorean triples: Baudhayana listed several sets of numbers satisfying a² + b² = c²
  • Approximation of √2: Accurate to 5 decimal places — an extraordinary achievement for 800 BCE
  • Geometric constructions: Methods for constructing squares equal in area to rectangles, circles equal in area to squares, and other transformations
  • Irrational numbers: Implicit awareness that √2 cannot be expressed as a simple fraction
💡 Remember: Baudhayana (~800 BCE) came approximately 300 years before Pythagoras (~500 BCE). When you see "Baudhayana–Pythagoras Theorem" in your NCERT textbook, recall that India contributed this knowledge to the world long before it was formalised in Greece.
📐 Statement of the Theorem
🔐 The Baudhayana–Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (called legs or perpendicular sides).

a² + b² = c² where c is the hypotenuse and a, b are the two legs of the right triangle

Right Triangle with Squares on Each Side

a² b² c² = a² + b² a b c (hypotenuse) A B C

In right triangle ABC with the right angle at B: AB² + BC² = AC²

📚 Key Terminology
Term Definition
Right-angled triangle A triangle in which one angle is exactly 90°
Hypotenuse The longest side of a right triangle, opposite the right angle
Legs (Perpendicular sides) The two sides that form the right angle
Pythagorean triple A set of three positive integers (a, b, c) such that a² + b² = c²
🔎 Understanding the Theorem Visually

The theorem has a beautiful visual interpretation. If you draw squares on each side of a right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

🔷 Example: 3-4-5 Triangle

Square on side 3: area = 3² = 9
Square on side 4: area = 4² = 16
Square on hypotenuse 5: area = 5² = 25
Check: 9 + 16 = 25 ✅

🔷 Example: 5-12-13 Triangle

Square on side 5: area = 5² = 25
Square on side 12: area = 12² = 144
Square on hypotenuse 13: area = 13² = 169
Check: 25 + 144 = 169 ✅
💡 Key Fact: The hypotenuse is always the longest side of a right triangle. It is always opposite to the 90° angle. When using the theorem, always identify the hypotenuse first!
🔢 Three Forms of the Formula

Depending on which side is unknown, the formula can be rearranged:

Finding Hypotenuse c

c = √(a² + b²)
Use when both legs are known.

Finding Leg a

a = √(c² − b²)
Use when hypotenuse and one leg are known.

Finding Leg b

b = √(c² − a²)
Use when hypotenuse and one leg are known.
🔬 Proof of the Theorem
📚 Proof by Rearrangement (Area Method)

This is one of the most elegant and visual proofs of the Baudhayana–Pythagoras theorem. There are over 400 known proofs, but this area-based proof is perhaps the most intuitive.

Proof: Using a Large Square with Side (a + b)

Step 1: Take a right triangle with legs a and b and hypotenuse c. Make four identical copies of this triangle.
Step 2: Arrange the four triangles inside a large square of side (a + b). Place them so their hypotenuses form a tilted square of side c in the centre.
Step 3: Calculate the area of the large square in two ways:
Method 1: Area = (a + b)² = a² + 2ab + b²
Method 2: Area = 4 triangles + inner square = 4 × (½ab) + c² = 2ab + c²
Step 4: Set the two expressions equal:
a² + 2ab + b² = 2ab + c²
Step 5: Subtract 2ab from both sides:
a² + b² = c² ✅

This completes the proof!
🔬 Bhaskara's Proof (Indian, 12th Century)

The great Indian mathematician Bhaskaracharya (Bhaskara II, 1114–1185 CE) gave one of the most famous proofs in history. He drew a diagram of a square with side c and inscribed a smaller square with side (a − b) inside it using four right triangles. He wrote just one word alongside his diagram:

"Behold!" Bhaskara's famous one-word proof — the diagram speaks for itself

In Bhaskara's proof, a square of side c is divided into 4 right triangles (each with area ½ab) and one inner square of side (a − b) with area (a − b)²:

Bhaskara's Calculation

Area of large square = c²
Area of 4 triangles + inner square = 4 × (½ab) + (a − b)² = 2ab + a² − 2ab + b² = a² + b²
Therefore: c² = a² + b² ✅
💡 Exam Tip: You do not need to memorise all proofs for the exam. The area method proof (using a large square of side a+b) is the most commonly asked. Understand it step by step and you can reproduce it easily.
🔄 Converse of the Theorem
📚 The Converse Statement

The converse of the Baudhayana–Pythagoras theorem is equally important:

If a² + b² = c² for the sides of a triangle, then the triangle is right-angled, and the right angle is opposite to the longest side c. Converse of the Baudhayana–Pythagoras Theorem

In simple words: if the square of the longest side equals the sum of the squares of the other two sides, then the triangle is right-angled.

🔎 How to Use the Converse
  1. Given three sides of a triangle, first identify the longest side (potential hypotenuse).
  2. Square all three sides.
  3. Check: Does (longest side)² = (side 1)² + (side 2)²?
  4. If yes, the triangle is right-angled.
  5. If no, the triangle is not right-angled.
✏️ Worked Examples

Example 1: Is a triangle with sides 6, 8, 10 a right triangle?

Step 1: Identify the longest side: 10 (potential hypotenuse).
Step 2: Check: 6² + 8² = 36 + 64 = 100. And 10² = 100.
Step 3: Since 6² + 8² = 10², the triangle is right-angled. ✅
The right angle is opposite the side of length 10.

Example 2: Is a triangle with sides 5, 7, 9 a right triangle?

Step 1: Longest side = 9. Check: 5² + 7² = 25 + 49 = 74.
Step 2: 9² = 81. Since 74 ≠ 81, this is not a right triangle. ❌

Example 3: Is a triangle with sides 7, 24, 25 a right triangle?

Step 1: Longest side = 25. Check: 7² + 24² = 49 + 576 = 625.
Step 2: 25² = 625. Since 7² + 24² = 25², the triangle is right-angled. ✅
📊 Acute vs Obtuse Triangle Check

The Baudhayana–Pythagoras theorem can also help classify triangles even when they are not right-angled:

✅ Right Triangle

a² + b² = c²
The triangle has a 90° angle.

🔶 Acute Triangle

a² + b² > c²
All angles are less than 90°.

🔴 Obtuse Triangle

a² + b² < c²
One angle is greater than 90°.
💡 Quick Check Mantra: Square the longest side. Square and add the other two.
• Equal? → Right triangle
• Sum is bigger? → Acute triangle
• Sum is smaller? → Obtuse triangle
🔢 Pythagorean Triples
📚 What Are Pythagorean Triples?

A Pythagorean triple is a set of three positive integers (a, b, c) such that a² + b² = c². These represent the sides of a right triangle where all side lengths are whole numbers.

📊 Common Pythagorean Triples
Triple (a, b, c) a² b² c² a² + b² = c²?
(3, 4, 5) 9 16 25 9 + 16 = 25 ✅
(5, 12, 13) 25 144 169 25 + 144 = 169 ✅
(8, 15, 17) 64 225 289 64 + 225 = 289 ✅
(7, 24, 25) 49 576 625 49 + 576 = 625 ✅
(9, 40, 41) 81 1600 1681 81 + 1600 = 1681 ✅
(11, 60, 61) 121 3600 3721 121 + 3600 = 3721 ✅
(20, 21, 29) 400 441 841 400 + 441 = 841 ✅
🔢 Generating Pythagorean Triples

Method 1: Scaling. If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

Scaling (3, 4, 5)

k=2: (6, 8, 10) ✅
k=3: (9, 12, 15) ✅
k=4: (12, 16, 20) ✅
k=5: (15, 20, 25) ✅

Scaling (5, 12, 13)

k=2: (10, 24, 26) ✅
k=3: (15, 36, 39) ✅
k=4: (20, 48, 52) ✅

Method 2: Using m and n (m > n > 0).

a = m² − n²,   b = 2mn,   c = m² + n² This generates all primitive Pythagorean triples when m and n have no common factor and are not both odd

Example: Generate a triple with m = 3, n = 2

a = m² − n² = 9 − 4 = 5
b = 2mn = 2 × 3 × 2 = 12
c = m² + n² = 9 + 4 = 13
Triple: (5, 12, 13) ✅ — Verify: 25 + 144 = 169 = 13²

Example: Generate a triple with m = 4, n = 1

a = 16 − 1 = 15,   b = 2 × 4 × 1 = 8,   c = 16 + 1 = 17
Triple: (8, 15, 17) ✅ — Verify: 64 + 225 = 289 = 17²
💡 Must-Know Triples for Exams: Memorise at least the first four: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Also remember that any multiple of a triple is also a triple. For example, (6,8,10) = 2 × (3,4,5).
✏️ Worked Examples (12 Examples)

Here are detailed worked examples covering all types of problems you will encounter in your exam.

Example 1: Find the hypotenuse of a right triangle with legs 6 cm and 8 cm.

Given: a = 6 cm, b = 8 cm, c = ?
Using: c² = a² + b² = 6² + 8² = 36 + 64 = 100
Answer: c = √100 = 10 cm

Example 2: The hypotenuse of a right triangle is 13 cm and one leg is 5 cm. Find the other leg.

Given: c = 13 cm, a = 5 cm, b = ?
Using: b² = c² − a² = 169 − 25 = 144
Answer: b = √144 = 12 cm

Example 3: A ladder 10 m long is placed against a wall. The foot of the ladder is 6 m from the wall. How high up the wall does the ladder reach?

Identify: The ladder is the hypotenuse (c = 10 m). The ground distance is one leg (a = 6 m). The wall height is the other leg (b = ?).
Using: b² = c² − a² = 100 − 36 = 64
Answer: b = √64 = 8 m. The ladder reaches 8 m up the wall.

Example 4: Find the diagonal of a rectangle with length 12 cm and breadth 5 cm.

Key insight: The diagonal of a rectangle divides it into two right triangles. The diagonal is the hypotenuse.
Using: d² = 12² + 5² = 144 + 25 = 169
Answer: d = √169 = 13 cm

Example 5: A ship sails 9 km due East and then 12 km due North. How far is the ship from its starting point?

Identify: East and North are perpendicular directions, forming a right triangle. The distance from start is the hypotenuse.
Using: d² = 9² + 12² = 81 + 144 = 225
Answer: d = √225 = 15 km

Example 6: Two poles of heights 6 m and 11 m stand 12 m apart. Find the distance between their tops.

Step 1: The difference in heights = 11 − 6 = 5 m (vertical leg). The horizontal distance = 12 m (horizontal leg).
Step 2: d² = 12² + 5² = 144 + 25 = 169
Answer: d = √169 = 13 m

Example 7: Check whether a triangle with sides 9, 12, and 15 is right-angled.

Longest side: 15. Check: 9² + 12² = 81 + 144 = 225.
15² = 225. Since 9² + 12² = 15², the triangle is right-angled. ✅
Note: This is 3 × (3, 4, 5).

Example 8: A 15 m long ladder reaches a window 12 m high. How far is the foot of the ladder from the wall? If the ladder is moved to reach a window 9 m high, how far will the foot be now?

Part 1: a² = 15² − 12² = 225 − 144 = 81 → a = 9 m
Part 2: a² = 15² − 9² = 225 − 81 = 144 → a = 12 m

Example 9: In a right triangle, if the two legs are in the ratio 3:4 and the hypotenuse is 20 cm, find the legs.

Let legs = 3x and 4x. Then: (3x)² + (4x)² = 20²
9x² + 16x² = 400 → 25x² = 400 → x² = 16 → x = 4
Answer: Legs = 3(4) = 12 cm and 4(4) = 16 cm

Example 10: The sides of a triangle are 20 cm, 21 cm, and 29 cm. Is it right-angled? If yes, find its area.

Check: Longest side = 29. 20² + 21² = 400 + 441 = 841. And 29² = 841. ✅ It is right-angled.
Area = ½ × 20 × 21 = 210 cm²

Example 11: Find the length of the altitude from the vertex to the base of an isosceles triangle with equal sides 13 cm and base 10 cm.

Key insight: In an isosceles triangle, the altitude from the apex to the base bisects the base. So the altitude creates two right triangles, each with hypotenuse 13 and base 5.
h² = 13² − 5² = 169 − 25 = 144
Answer: h = √144 = 12 cm

Example 12: A field is in the shape of a right triangle with the two perpendicular sides 48 m and 14 m. Find the length of the third side and the cost of fencing it at ₹50 per metre.

Third side (hypotenuse): c² = 48² + 14² = 2304 + 196 = 2500 → c = 50 m
Perimeter: 48 + 14 + 50 = 112 m
Cost of fencing: 112 × 50 = ₹5,600
🏗️ Real-World Applications

The Baudhayana–Pythagoras theorem is not just an abstract mathematical idea — it is used everywhere in the real world. Here are some important applications:

🚬 Ladders & Heights

Whenever a ladder is placed against a wall, it forms a right triangle with the wall and the ground. You can calculate how high the ladder reaches or how far its base should be from the wall.

🗻️ Navigation & Distance

Ships and aircraft use the theorem to calculate the shortest distance between two points when they have travelled in perpendicular directions (e.g., North then East).

🏗️ Construction & Architecture

Builders use the 3-4-5 rule to check if walls are perpendicular. They measure 3 units along one wall, 4 units along the other, and check if the diagonal is 5 units. If yes, the corner is exactly 90°.

📡 Diagonal of Screens

TV and phone screens are measured diagonally. If a TV screen is 48 inches wide and 36 inches tall, the diagonal (screen size) = √(48² + 36²) = √(2304 + 1296) = √3600 = 60 inches.

🗺️ Maps & GPS

Finding the straight-line distance between two points on a map uses the Pythagorean theorem (via the distance formula). Your phone's GPS does this calculation millions of times!

⚽️ Sports

In cricket or baseball, the distance the ball travels diagonally across the field is calculated using this theorem. A rectangular field's diagonal gives the maximum straight-line distance.

Real-World Example: The 3-4-5 Rule in Construction

Problem: A builder wants to check if the corner of a room is a perfect 90° angle.
Method: Measure 3 metres along one wall from the corner and mark it. Measure 4 metres along the other wall from the corner and mark it. Now measure the diagonal between the two marks.
Result: If the diagonal is exactly 5 metres, the corner is a perfect right angle! This works because (3, 4, 5) is a Pythagorean triple: 3² + 4² = 9 + 16 = 25 = 5².
💡 Real-Life Tip: The 3-4-5 method (or any scaled version like 6-8-10 or 30-40-50) is used by carpenters, masons, and builders all around the world to check right angles. It is the simplest practical application of the theorem!
🎯 Interactive: Explore the Theorem

Enter values for two sides of a right triangle and see the third side calculated instantly. Watch the theorem in action!

🔢 Right Triangle Calculator
Enter any two values and click "Calculate" to find the third side.
Enter any two sides and click Calculate to find the third!
a b c
🔬 Check if Three Sides Form a Right Triangle
🔄 Right Triangle Checker
Enter all three sides to check if they form a right triangle.
Enter three side lengths and click Check!
💡 Try These: Enter 3, 4, 5 in the calculator to verify. Then try 8, 15, 17. Now try 5, 7, 9 in the checker — is it a right triangle?
📊 Chapter Summary

📜 History

Baudhayana (~800 BCE) stated the theorem in the Sulba Sutras. Pythagoras (~500 BCE) later provided a formal proof. The NCERT textbook calls it the Baudhayana–Pythagoras Theorem.

📐 The Theorem

In a right triangle: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle).

🔬 The Proof

Can be proved using the area method: arrange 4 copies of the triangle in a square of side (a+b), compare areas to get a² + b² = c².

🔄 The Converse

If a² + b² = c² for three sides of a triangle, it must be right-angled (right angle opposite the longest side).

🔢 Pythagorean Triples

Integer triples satisfying the theorem: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple also works.

🏗️ Applications

Ladders, navigation, diagonals of rectangles, construction (3-4-5 rule), screen sizes, distance calculations.
📊 Quick Reference Table
To Find Formula When to Use
Hypotenuse c c = √(a² + b²) Both legs known
Leg a a = √(c² − b²) Hypotenuse and one leg known
Leg b b = √(c² − a²) Hypotenuse and one leg known
Right angle check Is a² + b² = c²? All three sides known — checking if right-angled
Diagonal of rectangle d = √(l² + w²) Length and width known
💡 Final Revision Mantra:
• Theorem: a² + b² = c² (in a right triangle)
• Hypotenuse: Always the longest side, opposite the right angle
• Converse: If a² + b² = c², the triangle is right-angled
• Key triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
• Heritage: Baudhayana (800 BCE) → Pythagoras (500 BCE)
• Steps: Identify hypotenuse → Apply formula → Solve → Check!
🧠 Multiple Choice Questions (10 MCQs)

Click on an option to see if your answer is correct. The correct option will turn green.

  • Q1. In a right-angled triangle, the side opposite to the right angle is called the:
    • a) Base
    • b) Perpendicular
    • c) Hypotenuse
    • d) Median
    ✅ Answer: (c) Hypotenuse — The side opposite the right angle is always the longest side, called the hypotenuse.
  • Q2. If the two legs of a right triangle are 3 cm and 4 cm, the hypotenuse is:
    • a) 7 cm
    • b) 5 cm
    • c) 12 cm
    • d) 25 cm
    ✅ Answer: (b) 5 cm — c = √(9 + 16) = √25 = 5 cm.
  • Q3. Which of the following is a Pythagorean triple?
    • a) (2, 3, 4)
    • b) (4, 5, 6)
    • c) (5, 12, 13)
    • d) (6, 7, 8)
    ✅ Answer: (c) (5, 12, 13) — 5² + 12² = 25 + 144 = 169 = 13².
  • Q4. The Baudhayana Sulba Sutra, which contains the earliest statement of the theorem, dates to approximately:
    • a) 200 BCE
    • b) 500 BCE
    • c) 800 BCE
    • d) 1200 CE
    ✅ Answer: (c) 800 BCE — Baudhayana composed the Sulba Sutras around 800 BCE, about 300 years before Pythagoras.
  • Q5. A ladder of length 17 m reaches a wall at a height of 15 m. The distance of the foot of the ladder from the wall is:
    • a) 2 m
    • b) 8 m
    • c) 10 m
    • d) 12 m
    ✅ Answer: (b) 8 m — d = √(17² − 15²) = √(289 − 225) = √64 = 8 m.
  • Q6. If the sides of a triangle are 6, 8, and 10, then the triangle is:
    • a) Equilateral
    • b) Isosceles
    • c) Right-angled
    • d) Obtuse
    ✅ Answer: (c) Right-angled — 6² + 8² = 36 + 64 = 100 = 10². This is a right triangle (it is 2 × (3,4,5)).
  • Q7. The diagonal of a rectangle with sides 9 cm and 40 cm is:
    • a) 49 cm
    • b) 31 cm
    • c) 41 cm
    • d) 45 cm
    ✅ Answer: (c) 41 cm — d = √(81 + 1600) = √1681 = 41 cm. Note: (9, 40, 41) is a Pythagorean triple!
  • Q8. In the converse of the Baudhayana–Pythagoras theorem, if a² + b² = c², then the triangle is:
    • a) Equilateral
    • b) Isosceles
    • c) Obtuse-angled
    • d) Right-angled
    ✅ Answer: (d) Right-angled — The converse states that if the square of the longest side equals the sum of squares of the other two, the triangle must be right-angled.
  • Q9. If (k, 24, 25) is a Pythagorean triple, then k equals:
    • a) 7
    • b) 10
    • c) 12
    • d) 15
    ✅ Answer: (a) 7 — k² + 576 = 625, so k² = 49, k = 7.
  • Q10. A ship travels 24 km due North and then 10 km due East. Its distance from the starting point is:
    • a) 34 km
    • b) 26 km
    • c) 14 km
    • d) 20 km
    ✅ Answer: (b) 26 km — d = √(576 + 100) = √676 = 26 km.
✍️ Short Answer Questions (10)
  • Q1. State the Baudhayana–Pythagoras theorem.
    In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If the legs are a and b and the hypotenuse is c, then a² + b² = c².
  • Q2. Who first described the theorem, and when?
    Baudhayana, an ancient Indian mathematician, first described the theorem around 800 BCE in the Baudhayana Sulba Sutra. This was approximately 300 years before the Greek mathematician Pythagoras (c. 570–495 BCE).
  • Q3. Find the hypotenuse of a right triangle with legs 5 cm and 12 cm.
    c² = 5² + 12² = 25 + 144 = 169. Therefore, c = √169 = 13 cm.
  • Q4. The hypotenuse of a right triangle is 25 cm and one leg is 7 cm. Find the other leg.
    b² = 25² − 7² = 625 − 49 = 576. Therefore, b = √576 = 24 cm.
  • Q5. State the converse of the Baudhayana–Pythagoras theorem.
    If in a triangle, the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is right-angled, and the right angle is opposite the longest side.
  • Q6. Check whether a triangle with sides 11, 60, and 61 is right-angled.
    Longest side = 61. Check: 11² + 60² = 121 + 3600 = 3721. And 61² = 3721. Since 11² + 60² = 61², the triangle is right-angled. ✅
  • Q7. Write three Pythagorean triples.
    (3, 4, 5): 9 + 16 = 25 ✅
    (5, 12, 13): 25 + 144 = 169 ✅
    (8, 15, 17): 64 + 225 = 289 ✅
  • Q8. A pole 24 m tall casts a shadow 7 m long. Find the distance from the top of the pole to the tip of the shadow.
    The pole and its shadow form a right angle (pole is vertical, shadow is horizontal). Distance = √(24² + 7²) = √(576 + 49) = √625 = 25 m.
  • Q9. Find the diagonal of a square whose side is 10 cm.
    The diagonal of a square forms a right triangle with two sides. d² = 10² + 10² = 100 + 100 = 200. Therefore, d = √200 = 10√2 ≈ 14.14 cm.
  • Q10. If (3k, 4k, 5k) is a Pythagorean triple for any positive integer k, verify for k = 3.
    For k = 3: the triple is (9, 12, 15). Check: 9² + 12² = 81 + 144 = 225. And 15² = 225. Since 225 = 225, it is a valid Pythagorean triple. ✅
📖 Long Answer Questions (5)
Q1. Prove the Baudhayana–Pythagoras theorem using the area method. Use a large square of side (a + b).
Proof:
Consider a right triangle with legs a and b and hypotenuse c. Make 4 identical copies.

Step 1: Arrange the 4 triangles inside a large square of side (a + b). Place them so their hypotenuses face inward, forming a tilted square of side c in the centre.

Step 2: Area of the large square = (a + b)² = a² + 2ab + b²

Step 3: The same large square is also composed of 4 right triangles + the inner square:
= 4 × (½ab) + c²
= 2ab + c²

Step 4: Equating the two expressions:
a² + 2ab + b² = 2ab + c²

Step 5: Subtracting 2ab from both sides:
a² + b² = c²

Hence proved. ✅
Q2. Two poles of heights 9 m and 16 m stand on a flat ground 12 m apart. Find the distance between their tops. Also, a wire is stretched from the top of the shorter pole to the top of the taller pole. Find the length of the wire.
Step 1: The difference in heights = 16 − 9 = 7 m.

Step 2: The tops of the poles and the horizontal form a right triangle with vertical side = 7 m and horizontal side = 12 m.

Step 3: Distance between tops = √(12² + 7²) = √(144 + 49) = √193

Step 4: √193 ≈ 13.89 m

The length of the wire stretched between the tops is also √193 ≈ 13.89 m, as the wire follows the straight-line distance between the two tops.
Q3. A man walks 40 m due East from a point A and reaches point B. He then turns and walks 30 m due North to reach point C. Find: (i) the distance AC, (ii) if he walks from C, 20 m due West to point D, find the distance AD.
(i) Finding AC:
AB = 40 m (East), BC = 30 m (North). ABC is a right triangle at B.
AC = √(40² + 30²) = √(1600 + 900) = √2500 = 50 m

(ii) Finding AD:
D is 20 m West of C. So D is at (40 − 20, 30) = (20, 30) relative to A.
AD = √(20² + 30²) = √(400 + 900) = √1300 = 10√13 ≈ 36.06 m
Q4. In an isosceles triangle ABC, AB = AC = 17 cm and BC = 16 cm. Find the altitude from A to BC and hence find the area of the triangle.
Step 1: Let the altitude from A to BC meet BC at point D. Since the triangle is isosceles (AB = AC), D is the midpoint of BC.
BD = DC = 16/2 = 8 cm

Step 2: In right triangle ABD:
AD² = AB² − BD² = 17² − 8² = 289 − 64 = 225
AD = √225 = 15 cm

Step 3: Area of triangle ABC = ½ × BC × AD = ½ × 16 × 15 = 120 cm²
Q5. Explain the historical significance of the Baudhayana Sulba Sutra in the context of the Pythagorean theorem. How did ancient Indian mathematicians contribute to the development of this fundamental theorem? Also verify whether (20, 21, 29) is a Pythagorean triple.
Historical Significance:
The Baudhayana Sulba Sutra (c. 800 BCE) is one of the oldest mathematical texts in the world. It was composed by the Vedic scholar Baudhayana as part of the Sulba Sutras — manuals for constructing geometrically precise fire altars for Vedic rituals.

In verse 1.48, Baudhayana states: "The diagonal of a rectangle produces both areas which its length and breadth produce separately." This is the earliest known statement of the theorem that the square on the diagonal equals the sum of squares on the two sides.

Indian Contributions:
• Baudhayana (~800 BCE): First statement of the theorem, approximation of √2
• Apastamba (~600 BCE): Listed several Pythagorean triples
• Brahmagupta (598–668 CE): Rules for zero and negative numbers, essential for algebraic development
• Bhaskaracharya (1114–1185 CE): Famous one-word proof ("Behold!") using area decomposition

These contributions predate Greek mathematics and demonstrate India's foundational role in the development of geometry.

Verification of (20, 21, 29):
20² + 21² = 400 + 441 = 841
29² = 841
Since 20² + 21² = 29², (20, 21, 29) is a Pythagorean triple. ✅
🌟 Fun Facts & Did You Know?

🇮🇳 India Was First!

Baudhayana stated the theorem around 800 BCE — roughly 300 years before Pythagoras was born. The NCERT textbook recognises this by calling it the Baudhayana–Pythagoras Theorem.

🔢 Over 400 Proofs!

The Pythagorean theorem has more known proofs than any other theorem in mathematics — over 400! Even US President James Garfield published a proof in 1876.

📜 Rope Geometry

"Sulba" means "rope" in Sanskrit. Ancient Indian priests used ropes of precise lengths to construct perfectly shaped altars. A rope with 12 equally spaced knots could make a 3-4-5 triangle!

💫 Bhaskara's "Behold!"

The great Indian mathematician Bhaskaracharya (12th century) gave one of history's most elegant proofs, accompanied by just one word: "Behold!" The diagram was so clear it needed no explanation.

🏗️ Ancient Builders

Egyptian pyramid builders used ropes with knots to create right angles using the 3-4-5 method, over 4000 years ago. The Pyramids of Giza demonstrate perfect right angles.

🚀 Into Space

NASA and ISRO use the 3D version of the Pythagorean theorem (d = √(x² + y² + z²)) to calculate distances between spacecraft and planets. The Chandrayaan mission used this extensively!

🎨 Spiralling Squares

If you draw right triangles with legs of 1 unit each, then use the hypotenuse of one as the leg of the next, you create a beautiful spiral called the "Spiral of Theodorus" or the "Square Root Spiral."

💻 Digital Screens

When you see "55-inch TV," that 55 inches is the diagonal measurement. It is calculated using the Pythagorean theorem from the width and height of the screen!
💡 Think About It: The Baudhayana–Pythagoras theorem connects the ancient Vedic fire altars of India to modern space missions. A mathematical truth discovered over 2800 years ago is still used every day in construction, navigation, gaming, and space exploration. That is the timeless power of mathematics!
🃏 Quick Revision Flashcards

Use these flashcards for last-minute revision before your exam. Read the question, try to answer mentally, then check!

Q: State the theorem.

A: In a right triangle, a² + b² = c², where c is the hypotenuse.

Q: Who stated it first?

A: Baudhayana (~800 BCE) in the Sulba Sutras.

Q: What is a Pythagorean triple?

A: Three positive integers (a, b, c) satisfying a² + b² = c². E.g., (3, 4, 5).

Q: Hypotenuse of 5, 12?

A: c = √(25 + 144) = √169 = 13.

Q: Is (7, 24, 25) a Pythagorean triple?

A: 49 + 576 = 625 = 25². Yes! ✅

Q: How to check for right angle?

A: If a² + b² = c² (converse), the triangle is right-angled.

Q: Diagonal of 8 × 6 rectangle?

A: d = √(64 + 36) = √100 = 10.

Q: What does "Sulba" mean?

A: Rope. The Sulba Sutras were guides for rope-based geometric constructions.

Q: Is 6, 8, 11 a right triangle?

A: 36 + 64 = 100 ≠ 121. No! ❌

Q: Name 4 Pythagorean triples.

A: (3,4,5), (5,12,13), (8,15,17), (7,24,25).

Q: Missing leg if c=10, a=6?

A: b = √(100 − 36) = √64 = 8.

Q: Who gave the "Behold!" proof?

A: Bhaskaracharya (Bhaskara II, 12th century).
💡 Final Revision Mantra:
• Theorem: a² + b² = c² (c is always the hypotenuse — the longest side)
• Converse: If the equation holds, the triangle is right-angled
• Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples)
• Heritage: Baudhayana (800 BCE) came before Pythagoras (500 BCE)
• Steps: Identify the right angle → Find the hypotenuse → Apply formula → Solve

You've got this! Go ace that exam! 💪
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