Pythagoras` Theorem

Let`s face it, we must have all heard about Pythagoras and his famous theorem. Throughout our school years, many teachers constantly drill this renowned theorem into our brains without actually telling us where it comes from. This article, we will see one of its many proofs.

Pythagoras` Theorem

Given a right-angled triangle, let us label the hypotenuse (the longest side opposite the right-angle) `c` and the two shorter sides `a` and `b` as shown below.

Then c = a + b .

(Note that this rule only works with right-angled triangles)

Let us have a look at this quick example.

Here, we have a triangle whose two shorter sides are 3 and 4. We wish to find the length of the hypotenuse. We are given the values of a and b. In fact, a = 3 and b = 4 (or you could switch them around). Indeed, it is c that we want to find.

Using the formula above, we have

c = 3 + 4

c = 9 + 16

c = 25

c = #873025 = 5

So c = 5.

As mathematicians, when we take the square root of a number, we must always consider both the positive and negative results (the square root of 25 is also -5 as -5 x -5 = 25). We cannot have c = -5 because we cannot have a negative length!

Notice also that all of our lengths of our right-angled triangle are whole numbers. Its lengths 3, 4 and 5 are known as Pythagorean triples. There are many other Pythagorean triples such as 5, 12 and 13, and 7, 24 and 25. I suggest you use Pythagoras` theorem to check that these are indeed Pythagorean triples -)

Proof of Pythagoras` Theorem

The main aim of this article is to share with you a proof of the theorem. In mathematics, once we make a statement, it is not necessarily true until we have proved that it is valid.

Let us consider the following:

We have two squares, one smaller than the other, and we have placed and rotated the smaller square inside the larger square to form four right-angled triangles. We have also labelled the lengths of these triangles a, b and c (just like in the theorem).

Notice that each length of the larger square is a + b. Let us call the area of the larger square Al. If we calculate the area of the larger square, we obtain Al = (a + b) x (a + b) = (a+b)(a+b) = a + 2ab + b

The last part has been obtained by expanding the brackets.

Now, let us consider the area of the smaller square (which we will call A_s). Each side has length c, so its area is A_s = c x c = c .

If we subtract the area of the smaller square from the area of the larger square, we obtain the total area of the four triangles. Let`s do that.

Area of four triangles = A_l - A_s = a + 2ab + b - c

We can now work out the area of the four triangles by calculating the area of one of the triangles and multiplying it by 4. The area of one of the triangles is x base x height = ab

Multiplying this by 4 gives Area of four triangles = 2ab.

We now have two expressions for the area of the four triangles. We can, therefore, equate them.

So, 2ab = a + 2ab + b - c

If we subtract 2ab from both sides, we get

0 = a + b - c

And, finally, adding c to both sides, we get

c = a + b .

And there you have it! We have proved Pythagoras` theorem.

There are many other proofs of the theorem, but at least you now know one of them :-)