The Baudhayana-Pythagoras Theorem

1Doubling a Square

Doubling a square here is a geometry job: build a square whose area is twice a given square. A right-isosceles diagonal gives a square on that diagonal with twice the area — the look this heading names, not a new kind of 2.

Drawing two squares side by side is not automatically the double-area square.

Figure. Baudhāyana’s doubling: the square built on the diagonal of a square has twice the area. The dashed diagonal of the unit square is the side of the square labelled 2.

How it works

1. Start with a square of side aArea a².
2. Use the diagonal as a new side if that is the construction you were taughtThe double job.
3. Check the new area is 2a²Doubling.

2Halving a Square

Halving a square is the reverse job: a square of half the area. Midpoints or a taught construction give a smaller square whose tiles are half. Half the side would quarter the area — that is not this half.

Cutting the side in half and calling the area half is the miss.

Figure. Halving is the reverse of doubling. Join the four midpoints of a square: the inner square has half the area. That inner square’s diagonal is a side of the outer square.

How it works

1. Start with area AThe given square.
2. Build a square of area A/2 as taughtThe half job.
3. Refuse “half the side” as half the areaSides square.

3Hypotenuse of an Isosceles Right Triangle

In an isosceles right triangle the two legs are equal. The hypotenuse is the side that faces the right angle. If each leg is 1, the square on the hypotenuse is 1+1=2, so the hypotenuse is √2 — a length you can name without a decimal hunt today.

Treating the hypotenuse as another 1 because the triangle “looks even” is a miss.

Figure. An isosceles right triangle has equal legs and a right angle between them. Baudhāyana then says the hypotenuse is a\sqrt{2}: the square on that edge is two copies of the square on a leg.

How it works

1. Mark the two equal legsThe isosceles right.
2. Write leg² + leg² = hyp²1+1=2.
3. Name hyp as √2 for unit legsThe heading’s length.

4Combining Two Different Squares

Two different squares can sit on the legs of a right triangle; the square on the hypotenuse equals those two combined. 3² + 4² = 5² is the number picture: 9+16=25.

Gluing two square papers at a corner is not the combine unless they are the leg-squares of one right triangle.

Figure. Two different squares add: the square on one leg plus the square on the other equals the square on the hypotenuse. Here 9 and 16 make 25, so the sides 3, 4, 5 close a right triangle.

How it works

1. Name the two leg-squaresa² and b².
2. Add those areasThe combine.
3. That sum is the hypotenuse-squarec².

5Right–Triangles Having Integer Sidelengths

Integer sidelengths that fit a² + b² = c² are a right triple: 5-12-13, 6-8-10 (a scaled 3-4-5). The heading is that they exist and you can check them, not a hunt for every triple today.

Any three whole numbers you like are not a right triangle.

How it works

1. Pick three wholes5, 12, 13.
2. Test a² + b² against c²25+144=169=13².
3. Pass or failA right triple or not.

6A Long-Standing Open Problem

A long-standing open problem here is a question people can still ask (as the lesson named it) — not a homework you must close. At this class you name that some questions stay open; you do not invent a proof.

Pretending you solved an open problem in one line is the miss.

Figure. Once 3-4-5 is found, more integer right triangles appear: 5-12-13, 8-15-17, … The long-standing question is the rule that generates all of them, not another lucky guess.

How it works

1. Hear the lesson’s open question as an open questionA named unknown.
2. Keep your job at the triples and the area-sum you can checkThis class.
3. Refuse a fake closeOpen means open.

7Further Applications of the Baudhāyana-Pythagoras Theorem

Further applications: a ladder against a wall, a diagonal of a rectangle, a path that turns 90°. Each is a² + b² = c² with a length you want. The application is a new story, same test.

A ladder photo without two lengths is not an application you can finish.

Figure. A ladder is a hypotenuse. If the wall drop is 12 and the foot is 5 from the wall, Baudhāyana gives the ladder 13 — because 25 + 144 = 169. Measure two sides; the third is forced.

How it works

1. Name the right angle and the two known sidesThe given.
2. Write a² + b² = c²The same theorem.
3. Solve the missing lengthThe application.

8A definition is a test you can run

The theorem is a test: in a right triangle, the hypotenuse-square equals the sum of the leg-squares. If the angle is not right, do not run this test. If you cannot name the right angle, you do not yet have the definition’s object.

A pretty 3-4-5 drawing that is not right-angled is a lying triple.

Figure. A definition is a test you run on the numbers, not a tidy-looking sketch. 36+64=100, so 6-8-10 earns a right angle. 36+64≠81, so 6-8-9 does not, however neat the drawing.

How it works

1. Find the right angleThe object.
2. Name legs and hypotenusec opposite 90°.
3. Run a² + b² = c²The test.