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Pythagoras' Theorem

Take any right-angled triangle. What's a right-angled triangle? It's simply a triangle with a right angle in it. Some people simply call them right triangles. Here's one:

Fig: A typical right-angled triangle. a, b and c stand for the lengths of each of the three sides.

The longest side in a right angled triangle, side c, is called the hypotenuse. It's always the longest because it has to bridge the gap between the ends of the two perpendicular (right angled) sides.

Pythagoras' theorem simply states that the length a squared plus b squared is equal to the length of the hypotenuse squared.

Mathematicallly, this can be written as:

a² + b² = c²

That's it. That's Pythagoras' theorem.

So, what's the point of Pythagoras' Theorem? Well, imagine you have a right-angled triangle where you know the lengths of two of the sides, and you want to know the length of the third side. This formula allows you to find the length of the third side easily. For example, imagine you know how long sides a and b are, and you want to know c. Just square the length of a, square the length of b, add the two squared lengths together (that gives you the length of the hypotenuse squared), then take the square root to get the length of the hypotenuse.

You can also rewrite Pythagoras' Theorem as a² = c² − b², which is useful if you know side lengths c and b but not a.

Not just triangles!

Admitedly, there aren't too many cases in the real world where you need to know one of the side lengths of a right angled triangle (there are notable exceptions, such as carpentry, engineering and of course maths exams).

But there are many cases of real world situations which can be represented by triangles. For instance, suppose you step out your front door and walk a distance, a, to the street; in a straight line, perpendicular to the street. When you get to the street, you turn left and walk a distance b, until your path is blocked by a ladder, standing upright at around 90° to the ground. Having nothing better to do, you climb the ladder. When you have climbed a distance c, you somehow manage to get stuck. You become quite hysterical. You are now a distance d from your front door. Fortunately, you have a cordless phone in your pocket. You can call for help. The cordless phone works by communicating with a base station, near the floor just inside your front door. If you're within close range of your front door, you can have a nice, clear, telephone conversation with the emergency services. Unfortunately, you happen to be an audiophile, and you value clear sound quality above all else. You will not even consider making the call if you think the sound quality will be inadequate for your finely tuned ears. So first, you want to work out the distance d, which will tell you exacly how far away you are from your front door.

You begin by squaring the distance from your front door to the street, a, as well as the distance along the street from your front gate to the base of the ladder, b. Then you add a² to b² and get the square of the distance from your front door to the base of the ladder. You could now take the square root of this to get a rough estimate of the distance to your front door, but you want to know precisely. You want to take account of the height of the ladder. So, you take the square of the distance from the base of the ladder to your front door, add it to the square of the height of the ladder, and get the square of the distance to your front door. You just need to take the square root of this to get the distance to your front door.

Mathematically, we could have written this as:

(a² + b²) + c² = d², or simply:

a² + b² + c² = d².

That's right, there may be no such thing as a three-dimensional triangle, but Pythagoras' theorem works in three dimensions! Now you can call for help, and rest assured with the knowlege that the crispness and clarity of your phone call will live up to the maximum potential your equipment allows.

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