3/26/15

Given a triangle ABC, we can "fold it in half" to get a new triangle: pick a vertex, e.g. A, and fold so that segments AB and AC line up. The same can be done from vertex B and vertex C, so there are three different ways to fold a triangle in half (Figure 1).

If the angles at vertices A, B and C are α, β, and γ respectively, with α ≤ β ≤ γ, what are the angles of the three possible triangles that can be obtained by folding ABC in half?

You can fold a 45-45-90 triangle at the right angle to obtain another 45-45-90 triangle. Describe all triangles that can be folded in half to get a similar triangle.

You can fold a 30-60-90 triangle to obtain a 30-30-120 triangle, then fold it again to get another 30-60-90 triangle. Describe all triangles that can be folded in half twice to get a similar triangle.

Prove that no matter how many times you fold a 40-60-80 triangle, you can never get another 40-60-80 triangle.

Hint: Solve the first bullet, and the rest will come out of the first result.

Level: 5