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Pythagoras – Finding C – Complete Lesson

Pythagoras – Finding C – Complete Lesson Preview the presentation to check ability-level, AFL questions, and the animations during demonstrations. It is recommended to delete slides/sections not needed for your class. Printing. To print handouts from slides -

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Pythagoras – Finding C – Complete Lesson

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  1. Pythagoras – Finding C – Complete Lesson Preview the presentation to check ability-level, AFL questions, and the animations during demonstrations. It is recommended to delete slides/sections not needed for your class.

  2. Printing To print handouts from slides - Select the slide from the left. Then click: File > Print > ‘Print Current Slide’ To print multiple slides - Click on a section title to highlight all those slides, or press ‘Ctrl’ at the same time as selecting slides to highlight more than one. Then click: File > Print > ‘Print Selection’ To print double-sided handouts - Highlight both slides before using ‘Print Selection’. Choose ‘Print on Both Sides’ and ‘Flip on Short Edge’.

  3. 15 Square +10 ÷2 Square -6 x3 √ 9 ?? +55 √ ÷3

  4. Use a calculator to find the answers. Answers Give your answers to 2 dp 42 = 2.852 = 5.32 + 5.12 = 3.52 = = 22 + 52 = = 52 + 2.32 = = = = = =

  5. Use a calculator to find the answers. Answers Give your answers to 2 dp 42 = 16 2.852 = 8.12 5.32 + 5.12 = 54.1 3.52 = 12.25 = ± 5.20 22 + 52 = 29 = ± 6.86 52 + 2.32 = 30.29 = ± 6.40 = +7.07, − 7.07 = ± 6.24 = But! You cannot have a square root of a negative number! = ± 3.16

  6. 13 April 2019 Pythagoras’ Theorem

  7. Give your answers to 2 dp. KNOWLEDGE CHECK 1) Find the length, . 6 cm 4 cm 3 cm 2) Find the length, . 4 cm 3) A football pitch is 97 m long and 42 m wide. Bill walks from one corner to another around the outside. Jane walks diagonally across the pitch. How much further does Bill walk?

  8. Give your answers to 2 dp. KNOWLEDGE CHECK 1) Find the length, . 6.71 cm 6 cm 4 cm 3 cm 2) Find the length, . 4 cm 14.42 cm 3) A football pitch is 97 m long and 42 m wide. Bill walks from one corner to another around the outside. Jane walks diagonally across the pitch. How much further does Bill walk? 33.30 m

  9. Jane’s cat is stuck on top of a wall! How long does a ladder need to be to reach the cat safely? 4 m

  10. Jane’s cat is stuck on top of a wall! How long does a ladder need to be to reach the cat safely? 4 m What shape has been made? What angle is here? Is the ladder longer than 4 m?

  11. Pythagoras of Samos c. 570 – c. 495 BC A Greek philosopher who taught students about religion and politics, and made mathematical discoveries.

  12. Pythagoras’ Theorem (only for right-angled triangles) 3cm Area = 9 cm2 5cm 5cm 3cm Area = 16 cm2 Area = 25 cm2 4cm 4cm

  13. Pythagoras’ Theorem (only for right-angled triangles) Area = 9 cm2 c a 5cm 3cm Area = 16 cm2 Area = 25 cm2 4cm b = c2 + b2 a2 a & b are the shorter sides. c is always the longest side. (the hypotenuse)

  14. Perigal’s Dissection (1891) A proof of Pythagoras’ Theorem. a c b a2 + b2 = c2

  15. Pythagoras’ Theorem Here is Triangle 1. It is a right-angled triangle with sides: 3 cm, 4 cm and 5 cm. (not to scale) We can draw a square on each side of the triangle. 5 cm 3 cm C 4 cm A What is the area of square A? ___________ What is the area of square B? ___________ What is the area of square C? ___________ B Continue the investigation… Triangle 2 13 cm 5 cm 12 cm 8 cm Triangle 3 15 cm 17 cm Triangle 5: sides of 12 cm, 35 cm and 37 cm. Conclusion: What is the relationship between the length of the sides of a right-angled triangle? Triangle 4 25 cm 7 cm 24 cm

  16. Pythagoras’ Theorem Here is Triangle 1. It is a right-angled triangle with sides: 3 cm, 4 cm and 5 cm. (not to scale) We can draw a square on each side of the triangle. 5 cm 3 cm C 4 cm A 9 cm2 What is the area of square A? ___________ What is the area of square B? ___________ What is the area of square C? ___________ 16 cm2 25 cm2 B Continue the investigation… Triangle 2 13 cm 5 cm 12 cm 8 cm Triangle 3 15 cm 17 cm Triangle 5: sides of 12 cm, 35 cm and 37 cm. Conclusion: What is the relationship between the length of the sides of a right-angled triangle? Answers Triangle 4 25 cm 7 cm 24 cm

  17. Pythagoras’ Theorem Pythagoras’ Theorem Here is Triangle 1. It is a right-angled triangle with sides: 3 cm, 4 cm and 5 cm. (not to scale) Here is Triangle 1. It is a right-angled triangle with sides: 3 cm, 4 cm and 5 cm. (not to scale) We can draw a square on each side of the triangle. We can draw a square on each side of the triangle. 5 cm 5 cm 3 cm 3 cm C C 4 cm 4 cm A A What is the area of square A? ___________ What is the area of square B? ___________ What is the area of square C? ___________ What is the area of square A? ___________ What is the area of square B? ___________ What is the area of square C? ___________ B B Continue the investigation… Continue the investigation… Triangle 2 Triangle 2 13 cm 13 cm 5 cm 5 cm 12 cm 12 cm 8 cm 8 cm Triangle 3 Triangle 3 15 cm 17 cm 15 cm 17 cm Triangle 5: sides of 12 cm, 35 cm and 37 cm. Triangle 5: sides of 12 cm, 35 cm and 37 cm. Conclusion: What is the relationship between the length of the sides of a right-angled triangle? Conclusion: What is the relationship between the length of the sides of a right-angled triangle? Triangle 4 Triangle 4 25 cm 25 cm 7 cm 7 cm 24 cm 24 cm

  18. R Write down the letter of each hypotenuse. (They are only in right-angled triangles!) Rearrange the letters to get an animal. K T H O E P D C P N W A G E A Q D P S T A G E S L P C O B F N M

  19. R Write down the letter of each hypotenuse. (They are only in right-angled triangles!) Rearrange the letters to get an animal. K T H O E P D C P N W A G E A Q D Elephant! P S T A G E S L P C O B F N M

  20. Example 1 1) Identify the hypotenuse and label the sides. What is the length of ? Give your answer to 2 d.p. a c 4 cm 2) Substitute the lengths into the formula. 5 cm b Not to scale 3) Find c2 a2 + b2 = c2 4) Square root to find c. 42 + 52 = 41 c2 = 41 We can leave the answer as a surd: which is exact, or round the number to 2 decimal places. c = cm = 6.40 (2dp)

  21. Example 2 What is the length of ? Give your answer to 2 d.p. a c 4 cm 7 cm b Not to scale a2 + b2 = c2 42 + 72 = 65 c2 = 65 c = cm = 8.06 (2dp)

  22. Example 2 Your Turn What is the length of ? Give your answer to 2 d.p. What is the length of ? Give your answer to 2 d.p. a a c c 4 cm 3 cm 7 cm 9 cm b b Not to scale Not to scale a2 + b2 = c2 a2 + b2 = c2 42 32 + 72 = 65 + 92 = 90 c2 = 65 c2 = 90 c = c = cm cm = 8.06 = 9.49 (2dp) (2dp)

  23. Example 2 Your Turn What is the length of ? Give your answer to 2 d.p. What is the length of ? Give your answer to 2 d.p. a a c c 4 cm 5 cm 7 cm 3 cm b Not to scale Not to scale b a2 + b2 = c2 a2 + b2 = c2 42 52 + 72 = 65 + 32 = 34 c2 = 65 c2 = 34 c = c = cm cm = 8.06 = 5.83 (2dp) (2dp)

  24. Example 2 Your Turn What is the length of ? Give your answer to 2 d.p. What is the length of ? Give your answer to 2 d.p. a 5 cm a c 8 cm b 4 cm 7 cm c b Not to scale Not to scale a2 + b2 = c2 a2 + b2 = c2 42 82 + 72 = 65 + 52 = 89 c2 = 65 c2 = 89 c = c = cm cm = 8.06 = 9.43 (2dp) (2dp)

  25. Example 2 Your Turn: Find the missing length () Give your answer to 2 dp What is the length of ? Give your answer to 2 d.p. A B a 5 cm 3 cm c 4 cm 7 cm 6 cm C 7 cm D 11.5 cm b Not to scale 4.5 cm a2 + b2 = c2 4 cm 3.5 cm 42 + 72 = 65 E c2 = 65 6.4 m c = cm = 8.06 6 m (2dp) 7 m

  26. Example 2 Your Turn: Find the missing length () Give your answer to 2 dp What is the length of ? Give your answer to 2 d.p. A B = 7.81 cm = 7.62 cm a 5 cm 3 cm c 4 cm 7 cm 6 cm C 7 cm D 11.5 cm b Not to scale = 5.70 cm 4.5 cm a2 + b2 = c2 = 12.18 cm 4 cm 3.5 cm 42 + 72 = 65 = 11.22 cm E c2 = 65 6.4 m c = cm = 8.06 6 m (2dp) 7 m

  27. Using Pythagoras’ Theorem to Find the Hypotenuse Remember! a2 + b2 = c2 1) Find the length of for each triangle. Answer to 2dp. a) b) c) 10 cm 4 cm 6 cm 6.5 cm 8 cm 9 cm = ______ = ______ = ______ d) e) f) 7.5 m 8.3 cm 6.1 cm 5 m 2.2 cm = ______ = ______ = ______ 2) Find the length of for each triangle but don’t use a calculator! Keep your answer as a surd instead. a) b) c) 5 cm 8 cm 4 cm 7 cm 5 cm 2 cm = ______ = ______ = ______ 3) Sketch a diagram to help answer these questions. Answer to 2sf. a) From his car, Mike walks 5 km north, and then 7 km east. How far away is he from his car now? ________ b) A boat sails directly south for 20 km. Then the boat sails west for 35 km. How far is the boat away from where it started? ________ c) A field is 100 m long and 75 m wide. Jack walks from one corner to another around the outside. Jane walks directly across the field. How much further does Jack walk? ________

  28. Using Pythagoras’ Theorem to Find the Hypotenuse Remember! a2 + b2 = c2 1) Find the length of for each triangle. Answer to 2dp. a) b) c) 10 cm 4 cm 6 cm 6.5 cm 8 cm 9 cm 8.94 cm 11.93 cm = ______ 10.82 cm = ______ = ______ d) e) f) 7.5 m 8.3 cm 6.1 cm 5 m 2.2 cm 9.01 m 6.48 cm 11.74 cm = ______ = ______ = ______ 2) Find the length of for each triangle but don’t use a calculator! Keep your answer as a surd instead. a) b) c) 5 cm 8 cm 4 cm 7 cm 5 cm 2 cm cm cm cm = ______ = ______ = ______ 3) Sketch a diagram to help answer these questions. Answer to 2sf. a) From his car, Mike walks 5 km north, and then 7 km east. How far away is he from his car now? 8.6 km ________ b) A boat sails directly south for 20 km. Then the boat sails west for 35 km. How far is the boat away from where it started? Answers 40 km ________ c) A field is 100 m long and 75 m wide. Jack walks from one corner to another around the outside. Jane walks directly across the field. How much further does Jack walk? 50 m ________

  29. Give your answers to 2 dp. KNOWLEDGE CHECK 1) Find the length, . 6 cm 4 cm 3 cm 2) Find the length, . 4 cm 3) A football pitch is 97 m long and 42 m wide. Bill walks from one corner to another around the outside. Jane walks diagonally across the pitch. How much further does Bill walk?

  30. Give your answers to 2 dp. KNOWLEDGE CHECK 1) Find the length, . 6.71 cm 6 cm 4 cm 3 cm 2) Find the length, . 4 cm 14.42 cm 3) A football pitch is 97 m long and 42 m wide. Bill walks from one corner to another around the outside. Jane walks diagonally across the pitch. How much further does Bill walk? 33.30 m

  31. What is special about these triangles? 4 m 12 m 3 m 5 m

  32. What is special about these triangles? 4 m 12 m 3 m 5 m These are called PythagoreanTriples because they are right-angled triangles with all integer lengths. Here are just a few others…

  33. Check your success! I can calculate the length of a hypotenuse using Pythagoras’ Theorem. I can use Pythagoras’ Theorem without a calculator. I can answer real-life questions using Pythagoras’ Theorem.

  34. Check your success! I can calculate the length of a hypotenuse using Pythagoras’ Theorem. I can use Pythagoras’ Theorem without a calculator. I can answer real-life questions using Pythagoras’ Theorem.

  35. Write a text message to a friend describing… What Pythagoras’ Theorem is.

  36. Questions? Comments? Suggestions? …or have you found a mistake!? Any feedback would be appreciated . Please feel free to email: tom@goteachmaths.co.uk

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