Combining two different squares

The idea

To combine two different squares, make a right-angled triangle whose perpendicular sides are their sidelengths; the square on its hypotenuse has area equal to the sum of the areas of the two squares.

Two squares, one square

Before, we joined two equal squares into one bigger square. Now take two different squares: one with side 3 and one with side 4.

Their areas are 3 × 3 = 9 and 4 × 4 = 16. We want one square whose area is exactly these two areas put together.

What area do you think that one square will have?

• 7
• 12
• 25

Baudhāyana's recipe

Press the button three times and follow the recipe. 1. Join the two squares at a corner. 2. Draw the right triangle whose perpendicular sides are the sides of the squares. 3. Build a square on its longest side, the hypotenuse.

The big square holds both

To combine two different squares, make a right-angled triangle whose perpendicular sides are the sides of the two squares. The square on its hypotenuse has area equal to the sum of the areas of the two squares.

For sides 3 and 4: 3 × 3 + 4 × 4 = 9 + 16 = 25. The big square has area 25, so its side is exactly 5, because 5 × 5 = 25.

Make a and b equal and you get the square on the diagonal from the last lessons: 3 × 3 + 3 × 3 = 18.

Notes

To combine two squares of sides a and b: make a right triangle with perpendicular sides a and b. The square on the hypotenuse has area a × a + b × b. For 3 and 4: 9 + 16 = 25.

Check yourself

Two squares have sides 2 and 3. A right triangle has perpendicular sides 2 and 3. What is the area of the square on its hypotenuse?

Answer: 13

2 × 2 + 3 × 3 = 4 + 9 = 13.

In Baudhāyana's recipe, where do the sides of the two squares go?

Two squares have sides 6 and 8. What is the area of the square on the hypotenuse of the right triangle with perpendicular sides 6 and 8?

Answer: 100

6 × 6 + 8 × 8 = 36 + 64 = 100. So its side is 10, because 10 × 10 = 100.

Which two squares can be combined into one square of area 50?

• They are the two perpendicular sides of a right triangle. — correct. Yes! The triangle has perpendicular sides a and b. The square on its hypotenuse is the big square.
• They are the hypotenuse and one perpendicular side of the triangle.. The hypotenuse is a new, longer side. The two squares' sides are both perpendicular sides.
• They are added to give the side of the big square: for 3 and 4, the side is 7.. A side of 7 would give area 49. But 9 + 16 = 25, so the big square is smaller than that.
• sides 3 and 4. Their areas add to 9 + 16 = 25, only half of 50.
• sides 2 and 6. Their areas add to 4 + 36 = 40, which is less than 50.
• sides 5 and 5 — correct. Yes! 5 × 5 + 5 × 5 = 25 + 25 = 50. Two equal squares: the new square is on the diagonal.