The Pythagorean Theorem

Ontario 09 Applied (MFM1P)

Pythagorean Spiral (Investigation)

Lesson

- To practice finding the missing side length of the hypotenuse using the Pythagorean Theorem.
- To understand some ways the Pythagorean Theorem is connected with nature.

- Protractor
- Metric Ruler
- Pencil
- 1 sheet of Computer Paper
- Crayons, Markers, or Colored Pencils

- Using your protractor draw a right angle slightly down and left from the center of the paper. Make sure both sides of this angle are 1 inch long (track the side lengths on a separate piece of paper).
- Connect the two legs to create a right triangle.
- Use the Pythagorean Theorem to determine the length of the hypotenuse of the triangle you just created.
- Use your protractor to create a 1 inch 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 the Pythagorean Theorem.
- Now use your protractor to create a 1 inch line perpendicular to the second triangle’s hypotenuse.
- Draw in the triangle’s hypotenuse as you did for the second triangle.
- Use the Pythagorean Theorem to determine the length of the new triangle’s hypotenuse.
- Continue the pattern. Stop drawing triangles when you are about to overlap the start 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 Pythagorean Theorem and record them.
- Optional: Decorate the spiral using your choice of crayons, markers, or colored pencils.

- Which triangle had the longest hypotenuse? What was its length? Why did it have the longest hypotenuse?
- Where in nature have you seen this pattern before? Think of some examples.

Solve problems using the Pythagorean theorem, as required in applications (e.g., calculate the height of a cone, given the radius and the slant height, in order to determine the volume of the cone)