2. http://www.mathsisfun.com/fraction s.html Year 9 Mathematics Pythagoras Theorem

3. Learning Intentions To correctly label a right-angled triangle to understand the relationship between the sides of right angled-triangle to solve problems using Pythagoras’ Theorem

4. Areas Your teacher will give you a copy of this diagram. Cut out the diagram along the lines carefully. Not try rearranging the pieces from squares B and C on top of square A. What do you notice?

5. The Result! When Pythagoras did this he noticed that: The area of the square made along side A was the same as the sum of the areas on sides B and C. Pythagoras stated this as: The square on hypotenuse is equal to the sum of the squares on the other two sides. Be careful – this only works for right-angled triangles!

6. Dissections! Henry Perigal (1801 – 1898) created a visual demonstration of Pythagoras’ Theorem.

7. Where is the Hypotenuse Pythagoras called the longest side in the triangle the Hypotenuse. Hypotenuse

8. The Algebra Pythagoras created an equation to help him solve problems involving right angled triangles. He wrote: a 2 = b 2 + c 2 But make sure you label the sides correctly! a b c

9. Writing the Formula Write out Pythagoras’ Theorem for the following triangles:

10. How Do I Use It? Calculate the size of the hypotenuse in this diagram. x 2 = 15 2 + 8 2 =225 + 64 =289 x=  289 x =17 cm 15 cm 8 cm x

11. Do that again Calculate the size of the hypotenuse in this diagram. x 2 = 3.6 2 + 4.8 2 =12.96 + 23.04 =36 x=  36 x =6 cm 3.6 cm x cm 4.8

12. Not the Hypotenuse! You need to be careful if you are finding the length of the side that is not the hypotenuse. 15 2 = 9 2 + x 2 225=81 + x 2 144=x 2  144=x x =12 cm 9 cm 15 cm x

13. Pythagorean Triples There are some special right angles triangles. 3, 4, 5 In this triangle the sides are all whole numbers. We can also enlarge this triangle to: 6, 8, 10 or 9, 12, 15 Or reduce it to: 1.5, 2, 2.5 25 9 16

14. Ancient Egypt The Egyptians new about the 3, 4, 5 triangle. They were able to use this knowledge in the construction of pyramids, temples and other buildings to ensure a perfect right- angle at the corners. They were in fact using the what was to become Pythagoras’ Theorem, 1500 years later. Rope with 12 equally spaced knots.

15. Pythagorean Triples 5, 12, 13 We can also enlarge this triangle to: 10, 24, 26 or 15, 26, 39 Or reduce it to: 2.5, 6, 6.5 169 144 25

16. Pythagorean Triples 7, 24, 25 We can also enlarge this triangle to: 14, 48, 50 or 21, 72, 75 Or reduce it to: 3.5, 12, 12.5 Recognising these triples can help to find the answers without using a calculator! 625 576 49

17. Questions

18. Is it a Right Angled Triangle? Sometimes you are asked if a triangle is right angled. To do this you must use Pythagoras’ theorem and check if it is correct. 18 7 11 17 8 15

19. Applications A skier travels down a steep slope loosing a vertical height of 560m as he covers 720 horizontally. Calculate the actual distance he skis down the slope.

20. Applications A 12m ladder rests against the side of a house. The top of the ladder is 9.5m from the floor. How far is the base of the ladder from the house? 12 m 9.5 m L

22. Applications The diagram shows an Olympic BMX. The frame of the bike is made from lightweight aluminium. The frame is shown in green. Calculate the total length of tubing needed to make the frame. Give your answer ain centimetres to 3 significant figures. Aluminium tubing weight 800g per metre. Work out the total mass of the frame in kg, to 3 significant figures.

23. Additional Questions How long is the diagonal of a square with a side of 6m? A ship leaves port and sails directly to a lighthouse 20 km east and 15 km north. How far has it sailed? Calculate the height of an equilateral triangle with a side length of 6 cm.