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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
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.
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
So if we know the lengths of the other two sides of the triangle, we can calculate the length of the hypotenuse.
Here is an example:
In this right-angled triangle, the hypotenuse is c.
x
y
z
p
q
r
In this right-angled triangle, the hypotenuse is _________.
In this right-angled triangle, the hypotenuse is _________.
Calculate the length of the hypotenuse of this right-angled triangle:
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.
The formula we used above was 𝑐 2 = 𝑎 2 + 𝑏 2. Let’s give the side we don’t
know a name – let’s call it x. So we know 𝑐 = 58, 𝑎 = 33 and 𝑏 = x.
Use the formula and replace 𝑐, 𝑎 , and 𝑏 with 58, 33, and x.
582 = 33^2 + 𝑥 2.
Using the calculator, we obtain:
4.1cm
7.6cm
58mm
33mm
Now we can subtract 1 089 from 3 364 and 𝑥 2 must be equal to that answer:
so 𝑥 2 = 2 275
Now find the square root of 2 275 and that is the value of the unknown side.
That is 𝑥 = √ 2275 = 47.69696007…
And so the 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.
14 cm 25cm
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)
Calculate the length of the hypotenuse (page 4)
Calculate the length of the unknown side (page 5)
x
y
z p q
r
In this right-angled triangle, the hypotenuse is y.
In this right-angled triangle, the hypotenuse is p.
4.1cm
7.6cm
Let the length of the hypotenuse be 𝑐 units.
= 4.1^2 + 7.6^2
So the hypotenuse is approximately 8.6cm long.
14cm 25cm
252 = 142 + 𝑥^2
So the unknown side is approximately 20.7cm long.
