The Pythagorean Theorem says that for *any* right triangle with sides a, b and c we have
a² + b² = c². Don’t forget, ‘c’ must be the length of the longest side of the right
triangle - the hypotenuse. If a triangle is *not* a right triangle, then this doesn’t
work: a² + b² ≠ c². Luckily right triangles are very common, so the Pythagorean Theorem
is extremely useful.
So, do you *believe* this is true or do you *know* this is true? There is a difference.
True mathematical knowledge comes when you understand *why* something is true. In mathematics,
there is no hoping and believing and wishing. But there is lots of proving. A proof is a
logical argument showing something is true. Let’s now prove the Pythagorean Theorem.
There are many, many ways to prove the Pythagorean Theorem. We’ll talk about one of my favorite
proofs.
To begin, let’s pick a random right triangle and label the sides a, b and c. Again, be
sure you label the hypotenuse c. Next, make 4 copies of this triangle... We are now going
to arrange these four triangles to form a square. Before we do this, let’s label the
angles in these triangles. This will help us in a moment... Let’s call the small angle
α and the next largest angle β. The *largest* angle is the right angle which is 90°. One
more thing. The angles in a triangle always sum to 180 degrees, so α + β + 90 = 180.
If you subtract 90 from both sides you get α + β = 90°. Another way to say this is
α and β are complementary angles.
Let’s begin by arranging two of the triangles so these sides form a straight line... That
means this is a 180° angle... And since α + β = 90°, this angle must be a right angle!
Next, arrange the third triangle so we have a straight side here... Again, since α +
β = 90°, this is also a right angle. And using similar logic, we can put the fourth
triangle here to complete the square.
The length of the sides of this square are a+b, so the area of the big square is (a+b)².
There’s another way to find the area of this square. We can add up the area of the
4 triangles, and the small square in the middle... Remember, the area of a triangle is ½ base
times height. So the area of each triangle is ½ a (times) b. The area of the small square
is c². So if we add up the area of the 4 triangles and the small square, we get ½
a b + ½ a b + ½ a b + ½ a b + c². If you collect like terms you get 2 a b + c².
We computed the area of this square in two different ways. Both ways give the same answer,
since they measure the area of the same square. So (a+b)² = 2 a b + c²... If you multiply
out the left hand side you get a² + 2 a b + b² = 2 a b + c². We can cancel 2 a b from
both sides, which leaves us with a² + b² = c². This is the Pythagorean Theorem! We
proved that it’s true!
There’s a nice Latin phrase you can use to say you are finished with a math proof:
quod erat demonstrandum... This means “which was to be demonstrated.” If you’re in
a hurry, you can shorten it to “q.e.d.” It’s a nice way to high-five yourself for
truly understanding why something works.