Chapter 9: The Baudhāyana–Pythagoras Theorem

1. Doubling and Halving a Square

Geometric dissection shows that areas can be rearranged without changing their total area. A square constructed on the diagonal of a unit square has area 2, leading to a side length of √2.

2. Isosceles Right Triangle and √2

In a right isosceles triangle with legs 1 and 1, the hypotenuse satisfies c²=1²+1²=2, so c=√2. This gives a geometric meaning to √2.

3. Combining Two Squares

If right-triangle legs have lengths a and b, the areas of the squares on those legs are a² and b². The central geometric relationship is a²+b²=c², where c is the hypotenuse.

4. Baudhāyana Triples

Whole-number triples such as 3,4,5 satisfy 3²+4²=5². Scaling gives 6,8,10 and 9,12,15. Such triples are useful for recognising right triangles quickly.

5. Fermat’s Last Theorem

The chapter connects the Pythagorean relationship with the deeper question of whether xⁿ+yⁿ=zⁿ has positive whole-number solutions for n greater than 2. Fermat’s Last Theorem states that no such non-zero whole-number solutions exist for n>2.

6. Applications

For a right triangle with legs 6 and 8, c²=36+64=100, so c=10. Always identify the right angle and the hypotenuse before using the theorem.

🧠 MCQs

1. For a right triangle, the longest side is the: A) base B) hypotenuse C) altitude D) median
2. For sides 3,4,5: A) 3²+4²=5² B) 3+4=5² C) 3²+5²=4² D) none
3. √2 is the diagonal length of: A) unit square B) 2-unit square C) triangle only D) circle
4. If legs are 5 and 12, hypotenuse is: A) 11 B) 12 C) 13 D) 17

Answers: 1-B, 2-A, 3-A, 4-C.

📌 Case Study

A rectangular playground is 9 m by 12 m. A student wants to walk directly from one corner to the opposite corner. Find the distance and explain the theorem used.

🔥 HOTS

1. Show why 5,12,13 is a Pythagorean triple.
2. Explain geometrically why the square on the hypotenuse has area equal to the sum of the areas on the legs.
3. Find two different right triangles with whole-number sides.

✍️ Practice

1. State the Pythagorean relationship.
2. Find the hypotenuse of a right triangle with legs 7 and 24.
3. Determine whether 8,15,17 forms a right triangle.
4. Explain the significance of √2 in the chapter.

🏆 Mastery Checklist

• ☐ I can identify the hypotenuse.
• ☐ I can apply a²+b²=c² correctly.
• ☐ I can recognise Pythagorean triples.
• ☐ I understand the geometric meaning of the theorem.

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