# Pythagoras' Theorem Exercise

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

##### MenuLevel 1Level 2Level 3Level 4Level 5Level 6ExamThree DHelpMore...

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!

 1. The length of the diagonal of a square is 85m. How long are the sides of the square? Give your answer correct to one decimal place. m 2. The blue squares have sides of length 24mm and the red square has sides of length 35mm. Find the distance from A to B in centimetres correct to one decimal place. cm 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 153m and the width is 96m. Give your answer to the nearest metre. m 4. Find the length of a side of a rhombus whose diagonals are of length 17km and 20km. Give your answer in kilometers correct to one decimal place. km 5. An irregular quadrilateral ABCD has right angles at the opposite vertices A and C. Calculate the length of the side DA if AB=34.9cm, BC=36.3cm and CD=36.4cm. Give your answers in cm to one decimal place. cm 6. An aeroplane flies due north for 293km then changes direction and flies east for 391km. How far is it now in a straight line from its starting position. Give your answer to the nearest kilometre. km 7. A ship sails on a bearing of 045o for 292km then changes direction and sails on a bearing of 135o for 403km. Finally it then turns and sails for 194km 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 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 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. m 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 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 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. cm²
Check

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 6. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5

## 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.

## Transum.org

This web site contains over a thousand free mathematical activities for teachers and pupils. Click here to go to the main page which links to all of the resources available.

Please contact me if you have any suggestions or questions.

## More Activities:

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?

Comment recorded on the 3 October 'Starter of the Day' page by Fiona Bray, Cams Hill School:

"This is an excellent website. We all often use the starters as the pupils come in the door and get settled as we take the register."

Comment recorded on the 14 October 'Starter of the Day' page by Inger Kisby, Herts and Essex High School:

"Just a quick note to say that we use a lot of your starters. It is lovely to have so many different ideas to start a lesson with. Thank you very much and keep up the good work."

Each month a newsletter is published containing details of the new additions to the Transum website and a new puzzle of the month.

The newsletter is then duplicated as a podcast which is available on the major delivery networks. You can listen to the podcast while you are commuting, exercising or relaxing.

Transum breaking news is available on Twitter @Transum and if that's not enough there is also a Transum Facebook page.

#### Pentransum

Answer multiple choice questions about basic mathematical ideas. If you get a number of questions correct you will be invited to post a question of your own. The bank of questions grows larger every day.

## Answers

There are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer.

A Transum subscription unlocks the answers to the online exercises, quizzes and puzzles. It also provides the teacher with access to quality external links on each of the Transum Topic pages and the facility to add to the collection themselves.

Subscribers can manage class lists, lesson plans and assessment data in the Class Admin application and have access to reports of the Transum Trophies earned by class members.

If you would like to enjoy ad-free access to the thousands of Transum resources, receive our monthly newsletter, unlock the printable worksheets and see our Maths Lesson Finishers then sign up for a subscription now:

Subscribe

## Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

## Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.

## Teachers

If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.

It may be worth remembering that if Transum.org should go offline for whatever reason, there are mirror sites at Transum.com and Transum.info that contain most of the resources that are available here on Transum.org.

When planning to use technology in your lesson always have a plan B!

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.

For Students:

For All:

©1997-2022 WWW.TRANSUM.ORG

## Description of Levels

Close

Level 1 - Finding the hypotenuse

Level 2 - Finding a shorter side

Level 3 - Mixed questions

Level 4 - Pythagoras coordinates

Level 5 - Pythagoras' Theorem exercise

Level 6 - Pythagoras' Theorem harder exercise

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.

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.

## 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.

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.

Alternatively the sides of the right-angled 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 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

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.

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

Close