Question 15.3.26

In Question 11.2.14 we considered the shortest path from a starting point, some distance north of a river running east-west, to a camp, stopping at the river on the way to get water. Diagram: river 4

We want to know the shortest total distance, $s = s_1 + s_2$, where \begin{align*} s_1^2 &= x_1^2 + Y_1^2 & s_2^2 &= x_2^2 + Y_2^2 \end{align*} by Pythagoras’ theorem (Section 11.2: Pythagoras' Theorem). I’m using the convention of capital letters for constants and lowercase letters for variables – $X$, $Y_1$, and $Y_2$ are given, whereas $x_1$, $x_2$, $s$, $s_1$, and $s_2$ depend on choices we make.

Show that $s$ is minimised when \[\frac{x_1}{s_1} = \frac{x_2}{s_2}\] that is, the two triangles must be similar. Show hint

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