IBM Ponder This #082 · February 2005

Puzzle for February 2005

Here's a geometric puzzle. All three parts need to be solved.

Start with a right triangle T1 whose vertices are (0,0), (0,1), (1,1). The angle at (0,0) is /4. Use the hypotenuse of T1 as the leg of a new triangle T2, whose angle at (0,0) is /8, and which extends down towards the x-axis. Use the hypotenuse of T2 as the leg of a new triangle T3, whose angle at (0,0) is /16. Continue ad infinitum.

The angles at (0,0) form an infinite series which sums to /2, so the limit of the process is a line segment on the x-axis.

Part 1: How long is that segment?

Part 2: Find an equation in polar coordinates (r,) of a curve passing through all the vertices of the resulting triangles, except possibly (0,0).

Part 3: Find an equation for that curve in rectangular coordinates (x,y).

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The first 100 people who answer correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

Solution

Best opened after a real attempt

To be added.