CONNECT: Pythagoras’ Theorem

You may remember from school that Pythagoras’ Theorem has something to do with right-angled triangles. So, firstly, we had better look at a right-angled triangle because we need to identify the important parts of it.

Diagram retrieved February 5, 2013, from http://www.schoolatoz.nsw.edu.au/homework-and- study/maths/maths-a-to-z/-/maths_glossary/RId5/203/right+angled+triangle

We identify a right-angled triangle by the small right-angle mark at one vertex (angle) of the triangle. As shown in the diagram, that angle measures 90°, which is a right angle. Because all the angles in a triangle add to 180°, the other two angles must add to 90°, and this means that neither of them can be larger than 90°, so the right angle is the largest angle in the right-angled triangle.

This also means that the side opposite the right angle (that is, the side that does not form one of the arms of the angle) is the largest side in the triangle. It is given a special name: the hypotenuse.

Diagram retrieved February 5, 2013, from http://spmath87308.blogspot.com.au/2009_02_01_archive.html

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We often use letters to identify the sides of a triangle, such as a, b, c and so on. Here are some different right-angled triangles identified by lettering. See if you can identify the hypotenuse in each case. (The first one is done for you.) You can check your results with the solutions at the end of this resource.

x

In this right-angled triangle, the hypotenuse is ______. z y In this right-angled triangle, the hypotenuse is c.

r

p In this right-angled triangle, the hypotenuse is ______. q

Diagrams retrieved February 5, 2013, from http://www.madsci.org/posts/archives/2000- 09/969073185.Ot.r.html and http://www.ck12.org/user:YWtlZWxlckBhY2VsZnJlc25vLm9yZw../section/Classifying-Triangles/

Pythagoras’ Theorem has to do with the length of the hypotenuse and the other two sides in a right-angled triangle. (Although the theorem is identified by Pythagoras’ name, it is doubtful that he actually was the first to prove it.)

If the length of the hypotenuse in a right-angled triangle is c units and the other two sides are a units and b units long, then it can be proved that

= + 2 2 2 So if we know the lengths of 푐the other푎 two푏 sides of the triangle, we can calculate the length of the hypotenuse.

Here is an example:

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Hypotenuse

10.5 mm

12.7 mm

In this triangle, the hypotenuse is marked for us, so we will use the letter c to represent its length in mm. The other two sides are given to us and we can choose one of them to be a and the other one to be b. It doesn’t matter which is which as long as you have the hypotenuse correct.

Using Pythagoras’ Theorem, we know that

= + , so 2 2 2 푐 = 10푎 .5 +푏 12.7 2 2 2 You need푐 to use your calculator here and get

= 271.54 2 That answer푐 seems very large even for the longest side of the triangle! That is because we have found . We now need to “undo” the by finding the square root of our answer. That2 means that 2 푐 푐 = 271.54 which is 16.4784708… (which is much more sensible for the length of the hypotenuse of this triangle). 푐 √ So, the length of the hypotenuse is approximately 16.5mm.

Over the page there is one for you to try. You can check your result with the solution at the end.

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Calculate the length of the hypotenuse of this right-angled triangle:

4.1cm

7.6cm

But what if we know the hypotenuse, and one of the other sides? Can we find the length of the third side? Yes!

Here is an example. We can find the length of the side that is not marked.

58mm

33mm

The formula we used above was = + . Let’s give the side we don’t know a name – let’s call it x. So we2 know2 = 58,2 = 33 and = x. 푐 푎 푏 Use the formula and replace , , and with푐 58, 푎33, and x. 푏

= + 푐 푎 푏 2 2 2 58푐 =푎33 +푏 . 2 2 2 Using the calculator, we푥 obtain:

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3 364 = 1 089 + . 2

Now we can subtract 1푥 089 from 3 364 and must be equal to that answer: 2

so = 2 275 푥 2 Now find the푥 square root of 2 275 and that is the value of the unknown side.

That is = 2275 = 47.69696007…

And so 푥the length√ of the third side of our triangle is approximately 47.7mm. (Look at the other lengths and make sure this makes sense – yes, it is not as long as the hypotenuse.)

Here is one for you to try. Find the length of the unknown side in the diagram. You can check your result with the solution at the end.

14cm 25cm

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Pythagorean Triads (or Triples)

You may have noticed that in the triangles we’ve looked at so far, even with those that have their lengths of sides as whole numbers or integers the 3rd side has always come out as a long decimal.

There are certain combinations of integers, however, that are called Pythagorean Triads (or Triples). These are sets of 3 whole numbers that can be substituted into Pythagoras’ formula and work.

Common examples are:

3, 4, 5. (We can check that 3 + 4 = 5 by looking at the numbers on the left of the =, and we get 92 + 162 and 2that’s equal to 25, and so is 5 on the right of the =, so this one works.) 2

8, 15, 17 is also a Pythagorean triad – you can check it in the same way.

5, 12, 13 is another Pythagorean triad, and so is 7, 24, 25 – again you can check both of these.

If you multiply any of these triads by the same number, you will also obtain a Pythagorean triad. For example, if we multiply all of 3, 4 and 5 by 2, we get 6, 8, 10. You can check if this combination of integers also fits Pythagorean formulas.

I have only listed a few of the more common Pythagorean triads. You can use them to make a right angled triangle: for example, if you cut three pieces of string so that they measure 3m, 4m and 5m, you can lay them out so that they form a right-angled triangle.

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If you need help with any of the Maths covered in this resource (or any other Maths topics), you can make an appointment with Learning Development through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11, or through your campus.

Solutions

Naming the hypotenuse (page 2)

x r

p z y q

In this right-angled triangle, the In this right-angled triangle, the

hypotenuse is y. hypotenuse is p.

Calculate the length of the hypotenuse (page 4)

Let the length of the hypotenuse be units. 4.1cm = + 푐 2 = 4.21 + 72 .6 푐 = 74.57푎 2 푏 2 = 74.57 7.6cm = 8.63539229 … So푐 the√ hypotenuse is approximately푐 8.6cm long.

Calculate the length of the unknown side (page 5) 25 = 14 +

2 2 2 625 = 196 + 푥

14cm 2 25cm 429 = 푥 2 = 푥429 = 20.71231518… So the unknown side is approximately푥 √ 20.7cm long.

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