Pythagoras' Theorem ExerciseA self marking exercise on the application of Pythagoras' Theorem. 
Here are some questions which can be answered using Pythagoras' Theorem. You can earn a trophy if you get at least 9 questions correct. Each time you finish a question click the 'Check' button lower down the page to see if you got it right! Diagrams are not to scale.
The last four questions were shared by Dr Colin Foster, Reader in Mathematics Education in the Mathematics Education Centre at Loughborough University, at his keynote address to the Mathematical Association as his some of his favourite "Pythagoras" tasks.
This is Pythagoras' Theorem Exercise level 7. You can also try:
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
InstructionsTry your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file. 



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Level 0  A 'whole number only' introductory set of questions
Level 1  Finding the hypotenuse
Level 2  Finding a shorter side
Level 3  Mixed questions
Level 4  Pythagoras coordinates
Level 5  Mixed exercise
Level 6  More than one triangle
Level 7  Harder exercise
Exam Style questions requiring an application of Pythagoras' Theorem and trigonometric ratios to find angles and lengths in rightangled triangles.
Three Dimensions  Three dimensional Pythagoras and trigonometry questions
More on this topic including lesson Starters, visual aids, investigations and selfmarking exercises.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.
See the National Curriculum page for links to related online activities and resources.
The area of the square on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares on the two shorter sides.
You may have learned the theorem using letters to stand for the lengths of the sides. The corners (vertices) of the rightangled triangle is labelled with capital (upper case) letters. The lengths of the sides opposite them are labelled with the corresponding small (lower case) letters.
Alternatively the sides of the rightangled triangle may me named using the capital letters of the two points they span.
As triangle can be labelled in many different ways it is probably best to remember the theorem by momorising the first diagram above.
To find the longest side (hypotenuse) of a rightangled triangle you square the two shorter sides, add together the results and then find the square root of this total.
To find a shorter side of a rightangled triangle you subtract the square of the other shorter side from the square of the hypotenuse and then find the square root of the answer.
AB^{2} = AC^{2}  BC^{2}
AB^{2} = 4.7^{2}  4.1^{2}
AB^{2} = 22.09  16.81
AB^{2} = 5.28
AB = √5.28
AB = 2.3m (to one decimal place)
The diagrams aren't always the same way round. They could be rotated by any angle.
The rightangled triangles could be long and thin or short and not so thin.
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Gertrudis Jobs,
Thursday, October 27, 2022
"Level 6  Question 9. I am standing in a rectangular hall, and my distances from three of the corners are 6 m, 9 m and 10 m. How far am I from the fourth corner? Give your answer correct to 3 significant figures.
wrong answer, answer is 12.0 (3 s.f).
[Transum: Thanks for your comment Gertrudis. Your answer is indeed correct if the order of measurements was different. You have made me realise that this question has three possible answers depending on the order of the measurements and which corner they refer to. This does make it a very interesting question. For the purposes of this exercise I have now added a diagram so the question only has one answer.]"