2D Applications of Pythagoras - Part 2 Slideshow 42, Mathematics Mr. Richard Sasaki

Objectives Understand facts about the Spiral of Theodorus Be able to draw the initial part of the spiral

The Spiral of Theodorus Today, we will learn how to draw the Spiral of Theodorus. We will look at facts about it after we try to draw it. This will be graded on design and whether it is correct or not. You need a ruler, compass, pencil and colouring pencils or other decorating items that you wish to use. In the end-of-term test, you will also need a ruler, compass and pencil. Try to figure out the lengths as you draw it!

The Spiral of Theodorus Firstly, we need to draw a 45-45-90 triangle. Have your paper landscape and from the centre (maybe slightly above), draw a 3 𝑐𝑚 line segment to the left Next, from the centre, draw another 3 𝑐𝑚 line segment perpendicular to the other Lastly connect the other vertices

The Spiral of Theodorus Next, we attach another triangle. From the hypotenuse of the original triangle, make a right angle and draw a 3 𝑐𝑚 line segment from the top vertex Connect the two remaining vertices together Consider the edges that are 3 𝑐𝑚 to be 1 unit. How long are the other two?

The Spiral of Theodorus Next, we attach the next triangle in the same way. From the hypotenuse of the new triangle, make a right angle and draw a 3 𝑐𝑚 line segment from the top vertex Connect the vertices on the left

The Spiral of Theodorus The rest is up to you, continue in the same way until you are to make a full rotation (16 triangles minimum). If you wish to continue further beyond that you may. After that, decorate it!

The Spiral of Theodorus 1 1 1 1 Let’s have a look at the mechanics of the spiral. 4 3 2 5 1 1 6 1 17 7 1 16 8 1 We know each of the edges shown are 1 unit. How about the others? 15 9 1 14 10 11 1 12 13 1 1 1 1 1 1

2 3 5 2 6 1 would be the 0th triangle’s hypotenuse. So kind of? The angles on the outside are approaching 180 𝑜 . 5 + 3 + 2 +3 2 𝐴 1+…+𝑛 = 1 2 + 2 2 +…+ 𝑛 2 No, the increase gets less as the triangle’s position increases. See the next slide.

Graph of 𝑦= 𝑥 2 As you can see, as 𝑥 increases, so does 𝑦 but the rate that 𝑦 increases gradually decreases. But this increase does not tend to a number of 𝑦 and will continue to increase forever. Note: Remember, we are talking about the area of the 𝑥th triangle in the Spiral of Theodorus!

Further Applications Even if the outer edges of the spiral differ as below, we can say… 𝑓 𝑑 𝑑 𝑒 𝑔 𝑒 𝑐 𝑏 𝑐 𝑏 𝑎 𝑎 𝑎 2 + 𝑏 2 = 𝑐 2 𝑎 2 + 𝑏 2 + 𝑑 2 + 𝑓 2 = 𝑔 2 𝑐 2 + 𝑑 2 = 𝑒 2 𝑎 2 + 𝑏 2 + 𝑑 2 = 𝑒 2 This works for any number of triangles!

12.5 = 5 2 2 20.75 = 83 2 𝐶𝐹=0.5 𝐶𝐸= 20 =2 5 2𝑥+15 𝑎 2 =𝑏+𝑐 (as 𝑏 and 𝑐 are consecutive)