1. Who wants to be a Millionaire? Pythagorean Triads.

2. Pythagorean Triads2 Video Clip courtesy of YouTube: Please view the video clip, from the TV quiz show “Who wants to be a Millionaire?”, at http://www.youtube.com/watch?v=BbX44YSsQ2I

3. Pythagorean Triads3 3 2 + 4 2 = 9 + 16 = 25 = 5 2

4. Pythagorean Triads4 Pythagoras’ Theorem Pythagoras’ Theorem states that, in a right- angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, a 2 + b 2 = c 2.

5. Pythagorean Triads5 Proof of Pythagoras’ Theorem

6. Pythagorean Triads6 The 3-4-5 Triangle It is not surprising that there are some right-angled triangles where all three sides are whole numbers called Pythagorean Triangles. The three whole number side- lengths are called a Pythagorean triple or triad. The most famous example is a = 3, b = 4 and c = 5, called "the 3-4-5 triangle". We can check it as follows: 3 2 + 4 2 = 9 + 16 = 25 = 5 2 so a 2 + b 2 = c 2. This triple was known to the Babylonians (who lived in the area of present-day Iraq and Iran) even as long as 5 000 years ago!

7. Pythagorean Triads7 More Pythagorean Triads Is the 3-4-5 the only Pythagorean Triad? No, because we can double the length of the sides of the 3-4-5 triangle and still have a right-angled triangle: its sides will be 6-8-10 and we can check that 10 2 = 6 2 + 8 2. Continuing this process by tripling 3-4-5 and quadrupling and so on, we have an infinite number of Pythagorean triads: 3 4 5 6 810 91215 121620 152025 182430... All of these will have the same shape (have the same angles) but differ in size - the mathematical term is that they are all similar triangles. If they were the same size but in different positions or orientations, the triangles are called congruent.

8. Pythagorean Triads8 Are there any other differently-shaped right- angled triangles with whole number sides? Yes; one is 5, 12, 13 & another is 7, 24, 25. We can check that they have right angles by using Pythagoras' Theorem that the squares of the two smaller sides sum to the square of the longest side. 5 2 + 12 2 = 25 + 144 = 169 = 13 2 7 2 + 24 2 = 49 + 576 = 625 = 25 2

9. Pythagorean Triads9 Which of the following is not a Pythagorean Triad? 3 : 4 : 59 : 40 : 4114 : 48 : 5020 : 48 : 52 5 : 12 : 1310 : 24 : 2615 : 20 : 2521 : 28 : 35 6 : 8 : 1011 : 60 : 6115 : 36 : 3921 : 72 : 75 7 : 24 : 2512 : 16 : 2017 : 90 : 9324 : 32 : 40 9 : 12 : 1513 : 84 : 8518 : 24 : 3025 : 60 : 65

10. Pythagorean Triads10 Which of the following is not a Pythagorean Triad? 3 : 4 : 59 : 40 : 4114 : 48 : 5020 : 48 : 52 5 : 12 : 1310 : 24 : 2615 : 20 : 2521 : 28 : 35 6 : 8 : 1011 : 60 : 6115 : 36 : 3921 : 72 : 75 7 : 24 : 2512 : 16 : 2017 : 90 : 9324 : 32 : 40 9 : 12 : 1513 : 84 : 8518 : 24 : 3025 : 60 : 65

11. Pythagorean Triads11 What is the total length a ? 3.6 4.8 6.0 6.4 7.2 9.6 10.0 a

12. Pythagorean Triads12 What is the total length a ? 3.6 4.8 6.0 6.4 7.2 9.6 10.0 a

13. Pythagorean Triads13 What is the length b ? 3.6 4.8 6.0 6.4 7.2 9.6 10.0 b

14. Pythagorean Triads14 What is the length b ? 3.6 4.8 6.0 6.4 7.2 9.6 10.0 b

15. Pythagorean Triads15 What is the length c ? 3.6 4.8 6.0 6.4 7.2 9.6 10.0 c

16. Pythagorean Triads16 What is the length c ? 3.6 4.8 6.0 6.4 7.2 9.6 10.0 c

17. Pythagorean Triads17 References / Sources: http://www.jaconline.com.au/mathsquestnsw/mq8nsw/investigations/05_pythag_triads.pdf http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html#345 http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/mcd/ M57501P.htm#a2