# Triangles and TrigonometryPythagoras’ Theorem

We have now reached an important point in geometry – being able to state and understand one of the most famous **Pythagoras’ Theorem**. It is named after the ancient Greek mathematician

**Pythagoras’ Theorem**

In any right-angled triangle, the square of the length of the **hypotenuse** (the side that lies opposite the right angle) is equal to the sum of the squares of the other two sides. In other words,

*The converse is also true: if the three sides in a triangle satisfy a ^{2} + b^{2} = c^{2}, then it must be *

Right angles are everywhere, and that’s why Pythagoras’ Theorem is so useful.

Here you can see a **6m** long ladder leaning on a wall. The bottom of the ladder is **1m** away from the wall. How far does it reach up the wall?

Notice that there is a right-angled triangle formed by the ladder, the wall and the ground. Using Pythagoras’ theorem, we get

Whenever you’ve got a right-angled triangle and know two of its sides, Pythagoras can help you find the third one.

## Proving Pythagoras’ Theorem

Pythagoras’ theorem was known to ancient Babylonians, Mesopotamians, Indians and Chinese – but Pythagoras may have been the first to find a formal, mathematical proof.

There are actually many different ways to prove Pythagoras’ theorem. Here you can see three different examples that each use a different strategy:

### Rearrangement

Have a look at the figure on the right. The square has side length *a* + *b*, and contains four right-angled triangles, as well as a smaller square of area

Now let’s rearrange the triangles in the square. The result still contains the four right-angles triangles, as well as two squares of size

Comparing the area of the red area before and after the rearrangement, we see that

This is the original proof that

### Algebra

Here we have the same figure as before, but this time we’ll use *algebra* rather than *rearrangement* to prove Pythagoras’ theorem.

The large square has side length

It consists of four triangles, each with an area of

If we combine all of that information, we have

And, once again, we get Pythagoras’ theorem.

### Similar Triangles

Here you can see another right-angled triangle. If we draw one of the altitudes, it splits the triangle into two smaller triangle. It also divides the hypotenuse *c* into two smaller parts which we’ll call x and y.

Let’s separate out the two smaller triangles, so that it’s clearer to see how they are related…

Both smaller triangles share one angle with the original triangle. They also all have one right angle. By the AA condition, all three triangles must be

Now we can use the equations we already know about similar polygons:

But remember that *c* = x + y. Therefore

Once more, we’ve proven Pythagoras’ theorem!

Much about Pythagoras’ life is unknown, and no original copies of his work have survived. He founded a religious cult, the *Pythagoreans*, that practiced a kind of “number worship”. They believed that all numbers have their own character, and followed a variety of other bizarre customs.

The Pythagoreans are credited with many mathematical discoveries, including finding the first

## Calculating Distances

One of the most important application of Pythagoras’ Theorem is for calculating distances.

On the right you can see two points in a coordinate system. We could measure their distance using a ruler, but that is not particularly accurate. Instead, let’s try using Pythagoras.

We can easily count the horizontal distance along the *x*-axis, and the vertical distance along the *y*-axis. If we draw those two lines, we get a right-angled triangle.

Using Pythagoras,

This method works for *any* two points:

**The Distance Formula**

If you are given two points with coordinates (

## Pythagorean Triples

As you moved the *d* ended up being a *all three sides* happens to be *whole numbers*.

One famous example is the 3-4-5 triangle. Since

The ancient Egyptians didn’t know about Pythagoras’ theorem, but they did know about the 3-4-5 triangle. When building the pyramids, they used knotted ropes of lengths 3, 4 and 5 to measure perfect right angles.

Three integers like this are called **Pythagorean Triples**

We can think of these triples as grid points in a coordinate systems. For a valid Pythagorean triples, the distance from the origin to the grid point has to be a whole number. Using the coordinate system below, can you find any other Pythagorean triples?

Do you notice any pattern in the distribution of these points?