On a rectangular table 5 units long and 2 units wide, $a$ ball is rolled from point $P$ at an angle of $45^{\circ}$ to $P Q$ and bounces off $S R$. The ball continues to bounce off the sides at $45^{\circ}$ until it reaches $S$. How many bounces of the ball are required?
(A) 9
(B) 8
(C) 7
(D) 5
(E) 4

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-147.jpg?height=254&width=466&top_left_y=2261&top_left_x=1317)
On a rectangular table 5 units long and 2 units wide, a ball is rolled from point $P$ at an angle of $45^{\circ}$ to $P Q$ and bounces off $S R$. The ball continues to bounce off the sides at $45^{\circ}$ until it reaches $S$. How many bounces of the ball are required?

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-302.jpg?height=279&width=520&top_left_y=218&top_left_x=1301)
(A) 9
(B) 8
(C) 7
(D) 5
(E) 4

## Solution

Since the ball bounces off the sides of the rectangular table at $45^{\circ}$, right-angled isosceles triangles are created as shown. The ball begins at point $P$ then bounces at points $A, B, C, D$, and $E$ before reaching $S$, for a total of 5 bounces.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-302.jpg?height=279&width=507&top_left_y=739&top_left_x=1297)

ANSWER: (D)