Combining two different squares

We can combine two copies of the same square into a bigger one, by building on the diagonal. What about two squares of different sizes?

Baudhāyana answers that too, in Verse 1.12: 'The area of the square produced by the diagonal is the sum of the areas of the squares produced by the two sides.'

In practice: make a right-angled triangle whose two perpendicular sides are the sides of your two squares. Then the square built on its hypotenuse has area equal to the two squares added together.

Why? Verse 2.1 gives the construction: mark a rectangle inside the larger square using a side of the smaller square; the diagonal of this rectangle is the side of a square whose area is the sum.

Follow it. Put the two squares, sides a and b, side by side. Inside the larger, mark off a rectangle a by b and draw its diagonal, giving a right triangle with legs a and b. Now draw three more copies of that triangle around the diagonal, making a four-sided figure.

That figure is a square. Its four sides are equal because all four triangles are congruent. And its angles are right angles: at each corner two angles meet, one of size x and one of size 90 − x, from the two triangles, so each corner is 90°.

Now count areas. The new square equals T + U + V; and by rearranging the same pieces it also equals V + W + X, which is exactly the two original squares. So the square on the hypotenuse has the area of both squares combined.

In letters, that's Baudhāyana's Theorem: if a right-angled triangle has sides a, b and hypotenuse c, then a² + b² = c². (It works only for right triangles, and the hypotenuse is always the side opposite the right angle.)

Baudhāyana was the first person in history to state it in this general, essentially modern form. It's also called the Pythagorean Theorem, after Pythagoras (c. 500 BCE), who admired and studied it and lived a couple of hundred years later. That's why the book uses the name Baudhāyana-Pythagoras Theorem, so everyone knows which theorem is meant.

Use it in a shape, too. A rhombus with diagonals 24 and 70: its diagonals cut each other in half at 90°, making right triangles with legs 12 and 35. So each side is √(12² + 35²) = √(144 + 1225) = √1369 = 37.