Chapter 14.2, Problem 25WE

To determine

To prove: that the total distance travelled by the ball equals the distance $BH"$ .

Explanation of Solution

Given information:

A reflection in one wall maps $H\text{ to }H\text{'}$ and to roll the ball from B to H, you aim for $H"$

Proof:

  $\begin{array}{l}\overline{OH\text{'}}=\overline{OH}\left[H\text{'}\text{is reflection of }H\right]\\ \overline{AH\text{'}}=\overline{AH"}\left[H"\text{is reflection of }H\right]\\ \text{therefore, }\overline{AO}+\overline{OH}=\overline{AH"}\end{array}$

Since total distance travelled $\overline{BA}+\overline{AO}+\overline{OH}$

Now, add $\overline{AB}$ on each side

$\begin{array}{l}\overline{AO}+\overline{OH}+\overline{AB}=\overline{AH"}+\overline{AB}\\ \overline{BA}+\overline{AO}+\overline{OH}=\overline{BH"}\left[\overline{AH"}+\overline{AB}=\overline{BH"}\right]\end{array}$

Therefore it is proved thatthe total distance travelled by the ball equals the distance $BH"$ .