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Pythagoras' Theorem Exercise

A self marking exercise on the application of Pythagoras' Theorem.

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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.

1. The length of the diagonal of a square is 71m. How long are the sides of the square? Give your answer correct to one decimal place.

m Correct Wrong

three squares

2. The blue squares have sides of length 28mm and the red square has sides of length 44mm. Find the distance from A to B in centimetres correct to one decimal place.

cm Correct Wrong

3. The sign says 'Keep off the grass'. Each day Michael has to get from one corner of the rectangular area of grass to the opposite corner. If the park keeper is looking he will walk along the edges but if the park keeper is not looking he will take the direct route, diagonally across the rectangle.

How much further does Michael walk on the days when the park keeper is looking? The length of the rectangular area of grass is 148m and the width is 110m. Give your answer to the nearest metre.

m Correct Wrong

4. Find the length of a side of a rhombus whose diagonals are of length 14km and 16km. Give your answer in kilometers correct to one decimal place.

km Correct Wrong

5. An irregular quadrilateral ABCD has right angles at the opposite vertices A and C. Calculate the length of the side DA if AB=33.7cm, BC=35.4cm and CD=37.4cm. Give your answers in cm to one decimal place.

cm Correct Wrong

6. An aeroplane flies due north for 291km then changes direction and flies east for 436km. How far is it now in a straight line from its starting position. Give your answer to the nearest kilometre.

km Correct Wrong

7. A ship sails on a bearing of 045o for 345km then changes direction and sails on a bearing of 135o for 383km. Finally it then turns and sails for 114km on a bearing of 225o. How far is it now in a straight line from its starting position. Give your answer to the nearest kilometre.

km Correct Wrong

8. One side of a right angled triangle is 10cm. The other two sides are both of length x. Calculate x to 3 significant figures.

cm Correct Wrong

9. I am standing in a rectangular hall, and my distances from three of the corners are 6 m, 9 m and 10 m as shown in the diagram below (not drawn to scale). How far am I from the fourth corner? Give your answer correct to 3 significant figures.

Rectangular Hall
m Correct Wrong

10. A wire 1 m long is lying flat along the ground, with its ends fixed. If its length is increased by 1 cm, but the ends are still fixed 1 m apart, how high up can the midpoint of the cable be raised before the cable becomes taut? Give your answer in centimetres correct to 3 significant figures.

cm Correct Wrong

11. What is the shortest distance from one corner of a 3cm x 5cm x 6cm cuboid to the opposite corner, travelling only along the surface of the cuboid?

cm Correct Wrong

12. The diagram shows two concentric circles and a line segment of length 3 which is a tangent to the smaller circle. Find the red shaded area correct to 3 significant figures.

Concentric circles
cm² Correct Wrong

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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

Instructions

Try 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.

Why am I learning this?

Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

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A version of the Play Your Cards Right TV programme. Calculate the probabilities of cards being higher or lower than the one shown. a fun way to practise applying probability and using fractions.

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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.]"

Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

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Description of Levels

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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

Pythagorean Probe - Levels 4 to 6 are quite a challenge.

Exam Style questions requiring an application of Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles.

Three Dimensions - Three dimensional Pythagoras and trigonometry questions

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Pythagoras' Theorem

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.

Pythagoras' Theorem

You may have learned the theorem using letters to stand for the lengths of the sides. The corners (vertices) of the right-angled triangle is labelled with capital (upper case) letters. The lengths of the sides opposite them are labelled with the corresponding small (lower case) letters.

Pythagoras' Theorem

Alternatively the sides of the right-angled triangle may me named using the capital letters of the two points they span.

Pythagoras' Theorem

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 right-angled 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 right-angled 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.

Example

Pythagoras Example

AB2 = AC2 - BC2
AB2 = 4.72 - 4.12
AB2 = 22.09 - 16.81
AB2 = 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.

Rotations

The right-angled triangles could be long and thin or short and not so thin.

Different proportions
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

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