Pythagoras' theorem

An inquiry challenge and its solution.

Will Eda finish as the winner, or as a pancake?

It is Friday evening and in 12 hours the footballers start training on the pitch.

Two snails make a bet about who reaches the mushroom on the other side of the pitch first. Both can race along at 10 m/h. The pitch measures 100 × 65 m.

Ota plays it safe – he goes around the pitch along the sides.

Eda takes the shortcut – he crawls straight across the pitch. But if he does not make it before the morning training…

Ota goes along the sides

He walks around two sides of the pitch.

Distance: 65 + 100 = 165 m
Time at a speed of 10 m/h: 165 ÷ 10 = 16.5 hours

Ota does not make it.

Eda goes across the pitch

He crawls straight along the diagonal, the hypotenuse of a right triangle. We use the Pythagorean theorem:

Length of the hypotenuse: c = √(65² + 100²)
The squares: 65² = 4,225 · 100² = 10,000
Sum: 14,225
Square root: √14,225 ≈ 119.3 m
Time: 119.3 ÷ 10 ≈ 11.9 hours

Eda arrives in less than 12 hours.

Who won?

Eda finished as the winner! And it was the Pythagorean theorem that let us work it out.

Keep exploring

1
Point to the three squares on the Koumio® board – one big one and two smaller ones.
2
Explore the triangle. Look at the labels of its sides: hypotenuse c, leg a, leg b. Which two sides form the right angle?
3
Fit all the red pieces into the area of the largest square, or
4
Stack all the red pieces into the areas of the two smaller squares.
5
Figure out what the three squares tell us about a right triangle.

What did you find out

Try to answer on your own first. The answer is hidden under the button.

The triangle in the middle is a right triangle. That means one of its angles is exactly 90°. The sides have names: a – leg, b – leg, c – hypotenuse. The hypotenuse is always the side opposite the right angle. And it is always the longest one.

Think about it: can a right triangle also have an obtuse angle? It cannot – because one angle already takes up 90°.

A square is built on each side of the triangle. On side a there is a square with area a², on side b a square with area b², on side c a square with area c².

And now comes the discovery. When you put the red pieces together, they either fill the one big square, or together they fill both smaller ones. The area of the big square is exactly the same as the sum of the areas of the two smaller ones. That is the Pythagorean theorem: a² + b² = c².

It holds for every right triangle. Just like Pythagoras, you have found out that in a right triangle the squares on the shorter sides (the legs) together have the same area as the square on the longest side (the hypotenuse).

What have you just discovered?

The area of the square on the hypotenuse is the same as the sum of the areas of the squares on the legs.
The Pythagorean theorem helps to work out real distances.

This is mathematics that works in the real world.