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Mathematics-Online problems:

Solution to the problem of the (previous) week

Problem:

Let a right triangle $ P_0P_1P$ have side lengths of 3, 4, and 5, with $ \overline{P_0P_1}$ being the shortest side of the triangle.
\includegraphics{A347_bild}

Determine the length $ l$ of the infinite polygon connecting the points $ P_0, P_1, P_2, P_3, \cdots$.


Answer:

$ l$ =

Solution:

$\displaystyle \frac{\vert\overline{P_1P_2}\vert}{\vert\overline{P_0P_1}\vert} = \frac{\sin{\sphericalangle (P_1P_0P_2)}}{\sin{(\pi/2)}} = \frac{4}{5} $

The similarity of the triangles implies that proportions remain equal.

Hence, the total length is

$\displaystyle l = 3\cdot \left(1 + \frac{4}{5} + \left(\frac{4}{5}\right)^2 + \cdots\right) = 15. $

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