Pythagoras

2. Pythagoras

3. The History of Pythagoras….. He lived from about 569BC to about 475BC and spent most his life in Italy. He was a Greek philosopher who made important developments in astronomy, mathematics and music. He never married, but had a group of followers called the Pythagoreans. He is best known for the Pythagorean Theorem which was known to the Babylonians 1000 years earlier but he may have been the first to prove it. Unfortunately, very little is known about Pythagoras because none of his writings have survived.

4. Pythagoras’ Theorem • “For any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.” a2 + b2 = h2 In a right angled triangle, the longest side is opposite the right angle and is called the hypotenuse (h). What is a right-angled triangle? What does hypotenuse mean? What does squaring a number mean? Sum of two numbers means?

5. Which is the hypotenuse – x, y or z x z x z y y x x y y z z

6. Proof: a² + b² = c²

7. Finding the length of the hypotenuse Examples 1: Find the value of x correct to 2dp. h2 = a2 + b2 h2 = 72 + 52 = 49 + 25 = 74 h = √74 = 8.60 (2dp) x 7 5

8. Example 2 Calculate the length of the hypotenuse 5² + 8² = h² 25 + 64 = h² 89 = h² h = h = 9.43 (2dp) 5m 8m

9. Riddle sheet Example 3: Find the length of the ladder correct to 2dp. h2 = a2 + b2 h2 = 42 + 62 = 16 + 36 h = √52 = 7.21m (2dp) Length of ladder = 7.21m (2dp) Exercise 18.2 pg 259

10. Testing out the Theory with a 3, 4, 5 triangle a2 + b2 = h2 Check: 3² + 4² = 9 + 16 = 25 =5² 5 3 4 Can you find another Pythagorean triple where a, b and c are less than 100? A Pythagorean Tripleis a triple of natural numbers (a,b,c) such that a2 + b2  = c2 e.g. 3, 4, 5

11. Do Now: Find the length of side c c 3.1m 3.1² + 8.5² = c² 9.61 + 72.25 = c² 81.86 = c² c = 9.045 (3dp) 8.5m

12. Finding a side which isn’t the hypotenuse Find the length of x to 2dp. h2 = a2 + b2 x2 + 22 = 52 x2 + 4= 25 x2 = 25 – 4 x2 = 21 x = √21 = 4.58 (2dp) x 5 2

13. Find the length x that a 10m ladder would go up the wall if the foot of the ladder is 2m from the wall. h2 = a2 + b2 x2 + 22 = 102 x2= 100 – 4 x = √96 = 9.80 (2dp) Diagonal distance = 9.80m (2dp)

14. Do Now 5252 =4502 +b2 73125 = b2 b = 270.42km

15. Applications • Draw a diagram of a right-angled triangle. • Put the measurements on the diagram, placing the unknown as x. • Solve for x leaving your answer in sentence form.

16. If you stand 4.2m from a 6.2m high tree and look diagonally at the top, how far is it from the top of your head to the top of the tree if you are 1.6m tall? Give your answer to 1dp. h2 = a2 + b2 x2 = 4.62 + 4.22 = 21.16 + 17.64 = 38.8 x = √38.8 = 6.2 (2dp) The diagonal distance is 6.2m (1dp). Riddles – What do two bullets have when they get married? - What did Lancelot say to the beautiful Ellen? - What happened at the milking contest?

17. Do Now: • Use Pythagoras’ Theorem to find the length of the side labelled x. x = 6m 1. 2. x 8m x 5.8cm x = 6.17cm 10m 2.1cm

18. An Air NZ plane takes off from Dunedin airport heading to Nelson. The plane climbs at an angle of 35° from the time it takes off and flies at this angle for 10km. Unfortunately after flying for 10km at an angle of 35° disaster strikes and the plane is struck by a huge lightning bolt, causing the plane to fall from the sky. If the plane was at a height less than 6.5km, then there will be survivors. But if the plane was higher than 6.5km when it was hit, then there will not be any survivors. I happened to be with a group of people directly under where the plane was hit by lightning and someone shouted out ‘Are there any mathematicians here, does anyone know how to do trigonometry to work out how high the plane was?’ Luckily I was there to use my trigonometry skills so I could work out how high the plane was when it hit and I was able to tell the people whether we needed to call an ambulance for the survivors or if we needed body bags.

19. Do Now: 1. 42.2 m (1dp) 2. For each of these triangles write down which side is the hypotenuse, opposite and adjacent. e p d f q r

20.  Labelling sides of triangles • Hypotenuse = the longest side opposite the right angle. • Opposite = opposite the marked angle. • Adjacent = side that joins the right angle to the marked angle. Hypotenuse (H) Adjacent (A) Opposite (O) Hypotenuse (H) Opposite (O)  Adjacent (A)

21. Sohcahtoa The three trig ratios are • sin = • cos = • tan = They allow you to find a side or angle of a right angled triangle if you know an angle or one side or 2 angles. Sometimes this is written in the form below

22. Quick Quiz 1. x 3 • What is the ratio for the sine of an angle? 9.5 (1dp) 9 x • What is the ratio for the cosine of an angle? 2. 14.89 (2dp) 3.5 15.3 • What is the ratio for the tangent of an angle? 3. Which is the hypotenuse, opposite, adjacent? f d e

23. Finding a side • When given a side length and an angle, you can not use Pythagoras’ Theorem. • Identify which side you have and which you need. • Decide which rule to use. • Substitute values and rearrange to solve.

24. Example Find the length of x. Have o and h so use soh Want o so cover o and get x = sin 60 x 14 = 12cm (nearest cm) x 14cm h o 60°

25. x 4cm 40 Find the length of x. Have o and h so use soh Want h so cover h and get x = = 6cm (nearest cm) h o

26. Find the length of x. Have a and h so use Cah Want a so cover a to get x = cos45 x 10 = 7cm (nearest cm) a 10 x 45 h

27. Find the length of x. Have a and o so use toa Want a so cover a to get x = = 9cm (nearest cm) a x 35 6cm o

28. Do Now: • A ladder, 5 metres long, leans against a wall and makes an angle of 70° with the ground. How far is: • The top of the ladder from the ground? • The foot of the ladder from the wall? (Hint: draw a diagram first!) a) 4.7m (1dp) b) 1.71m (2dp)

29. Ex 20.7 pg 294 - Riddle • Worksheet • Ex 20.3 pg 285 - Applications

30. Do Now • Find x Use soh o = sin 74 x 4 = 3.85km 2. Find y Use cah A = cos 74 x 4 = 1.10km 74° 4km y km x km 3. Draw the trig ratio triangles in your book

31. Finding an Angle When given a problem where you have to find the angle : • Identify which two sides are given. • Decide which rule to use (SOH-CAH-TOA). • Substitute values and solve.

32. Inverse sin, cos, tan • When we use the trig triangles to find the angle, we find sinθ, cosθ or tanθ. • So to find the angle θ, we need to use the inverse button (shift sin, cos or tan) • Find θ if • Sin θ = 0.5 30o • Cos θ = 0.688 46.5o • tan θ = 1.34 53.3o

33. Find the angle Have o and h so use soh Want angle so cover sin to get Sinθ = = 0.5 θ = sin-1 0.5 = 30o h 10 5 o  Shift sin

34. Find the angle Have a and h so use Cah Want angle so cover cos Cos θ = θ = cos-1 ( ) = 70.5o (1dp) 12 h a 4

35. 12 o Find the angle Have a and o so use Toa Want angle so cover tan tan θ = θ = tan-1 ( 3 ) = 71.6o (1dp) 4 a Ex 21.3 pg 297 Ex 21.4 pg 298

36. Find x • Find x Use soh o = sin 62 x 5 = 4.41km Find θ Sin θ = 0.4 23.6o Cos θ = 41.4o tan θ = 32o 62 5km x km

37. DO Now 6km Find the angle • Use soh • Sin-1 ( ) • = 30o Find the other side Could use Pythagoras, cos or tan! 10.4km 12km

38. A good game

39. Applications Find the height of the building Have o and a So use toa Want o so cover o to get Height = tan 39 x 46.0 = 37.3m

40. Find the angle of the road Have o and h So use soh Want angle so cover sin to get Sin θ = θ =sin-1 0.1767 = 10.18o • Ex 21.4 pg 298 • When finished get Riddle sheet