INVESTIGATION: The spiral of Theodorus

To investigate an interesting mathematical pattern
To consolidate the skill of finding the hypotenuse of a right-angled triangle using Pythagoras' theorem

Using your protractor draw a right angle slightly down and left from the centre of the paper. Make sure both sides of this angle are 1 cm long.

Connect the two legs to create a right triangle.
Use the Pythagoras theorem to determine the length of the hypotenuse of the triangle you just created.
Record the length of the hypotenuse in a table in your notebook such as the following.

Use your protractor to create a 1 cm line perpendicular to the first triangle’s hypotenuse.

Connect the lines to create the hypotenuse of a second right triangle. Notice the hypotenuse of the old triangle now acts as one of the legs of the new triangle.
Calculate the hypotenuse of the new triangle using Pythagoras' theorem.
Now use your protractor to create a 1 cm line perpendicular to the second triangle’s hypotenuse.
Draw in the triangle’s hypotenuse as you did for the second triangle.
Use Pythagoras' theorem to determine the length of the new triangle’s hypotenuse.
Continue the pattern. Stop drawing triangles when you are about to overlap the original triangle (do not overlap triangles). In the end, you should have 16 triangles. There will be a small gap between the first and last triangles. As you go, continue to find the length of each hypotenuse you create using the Pythagoras' theorem and record them.
Optional: Decorate the spiral using your choice of crayons, markers, or coloured pencils.

Where in nature have you seen this pattern before? Find some examples.
What pattern do the lengths of the hypotenuses follow? Why does this happen?
Which triangle had the longest hypotenuse? What was its length? Why did it have the longest hypotenuse?

apply Pythagoras’ theorem to solve problems for all side lengths using 𝑎^2 + 𝑏^2 = 𝑐^2