GCSE Maths: Pythagoras’ Theorem - 1 Formula That Changed Maths
Introduction
Whether you are doing Foundation GCSE Maths or Higher GCSE Maths, you are going to see one question involving Pythagoras’ Theorem. Some questions can be straightforward where you just need to work out a missing length. Others may require you to use the theorem more than once in order to obtain the answer.
Read the rest of this article and look at the examples, try them and then try them without looking at the method. Can you get the same answer?
Pythagoras' Theorem: What you need to know
This is a theorem that relates to right angled triangles only. You should be able to memorise the theorem and to be able to apply it in order to find the longest side of a triangle as well as one of the shorter sides of a triangle.
As well as this you could be given a scenario based question where a diagram will be useful and Pythagoras’ Theorem should be used.
Example
Take a look at the following question:
From the diagram it is clear that you need to determine the longest side. Using Pythagoras’ Theorem:
(Y Z)^2=(1.7)^2+(3.2)^2Now one thing that is very important when it comes to the calculation is to not to think that 1.72+3.22= 4.92. This is not the case.
What is on the right hand side of the equal side should be inputted carefully into a calculator as it is shown. Doing this will give 13.13
(Y Z)^2=13.13In order to determine the value of YZ the square root of 13.13 must now be taken, i.e. \sqrt{13.13}=3.62 \mathrm{~cm}
Example
Consider the following question:
From the diagram that is given in this question you will see that you are given the length of the longest side and one of the shorter side and it is the other shorter side that needs to be determined.
Again Pythagoras’ Theorem should be used and the information you have should be entered as shown:
(Q R)^2+8^2=16^2Now the calculation here is different to the calculation that was done in the first example.
You want to have (Q R)^2 on its own on the left hand side, so a simple algebraic technique needs to be applied where 82 is subtracted from both sides to give:
(Q R)^2=16^2-8^2Again it is important to remember that you cannot just add or subtract square numbers. What is written on the right hand side needs to be entered into a calculator as shown. This will then give:
(Q R)^2=192 \rightarrow Q R=\sqrt{192}=13.86 \mathrm{~cm}Question Practice
Try the following question on your own before looking at the solution.
Solution
So how did you get on? Hopefully you found the answer to be 5.8m.
In this question you are not given any diagram as an aide, there are just words and there is even no suggestion that you need to use Pythagoras’ Theorem. The best way to answer a question like this is to start off with a simple diagram.
You can see that a diagram has been drawn and what you can actually see is a right angled triangle where one of the shorter sides needs to be found. Let the unknown side be represented by \chi then:
6^2=1.5^2+x^2Subtracting 1.5^2 from both sides gives:
\begin{gathered} x^2=6^2-1.5^2=33.75 \\ \therefore x=\sqrt{33.75}=5.8 \mathrm{~m} \end{gathered}Hopefully you found the questions reasonable to answer by yourself but as mentioned there can be much more complicated questions and these tend to be combined with SOH CAH TOA and/or even the Sine and Cosine rules. These more complicated questions tend to be found on the GCSE Higher Paper.
Quite often the biggest mistake with Pythagoras is knowing when to do a subtraction or an addition.
If finding the “longest” side you square and “add” the two shorter sides.
If finding a “shorter” side then you square and “subtract” the two other sides.
Our 2 or 3 day GCSE Maths Revision Courses will help you overcome any misunderstandings that you may have as well as give you the confidence that you need to tackle any type of question. Courses strategically take place during the half term holidays so you have the perfect opportunity to focus on your revision and consolidate your understanding of any topics.
Whatever your goals if you need help getting those top grades then just complete the form and we will be in contact within 24 hours.