Cool Stuff to Make with Right Triangles

(Individual homework)

Construct a spiral of right triangles as follows.

1. Construct an isosceles right triangle. Call the length of each of its legs 1. What is the length of the hypoteneuse?
2. Use the hypoteneuse of the first triangle as one of the legs of the second triangle. Construct a perpendicular segment of length 1 at one end of the segment. Connect vertices to make another right triangle. What is the length of the hypoteneuse of the second triangle?
3. Continue to construct right triangles with one leg of length 1, and one leg the hypoteneuse of the previous triangle. Construct at least 6 triangles.
4. Write a formula for the length of the hypoteneuse of the nth triangle.
5. Make a conjecture about lengths that are constructible with straightedge and compass.

In this activity you will construct a puzzle consisting of 3 square pyramids.

All 3 pyramids are congruent; each fits inside a cube. Call the length of an edge of the cube 1.

The base of the cube, ABCD, is the base of the pyramid

The apex of the pyramid, E, is another vertex of the cube.

So the faces of the pyramid are a square (ABCD), and 4 triangles (ABE, BCE, CDE, ADE).

1. Find the lengths of the edges AE and CE, and prove that ADE and CDE are right triangles.
2. Find the length of edge BE and prove that ABE and CBE are right triangles. Which are the right angles in these triangles?
3. Make a net (flat pattern of connected polygons) for the pyramid.
   Cut out and fold 3 identical pyramids.
   Assemble the 3 pyramids to make a cube.
4. What is the volume of the cube? (Remember that the edge length is 1.)
5. What is the volume of one pyramid?

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