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_6ed09463f225f8ba1f07g-149.jpg?height=249&width=475&top_left_y=1670&top_left_x=1302)
(A) 9
(B) 8
(C) 7
(D) 5
(E) 4
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_faa9db06d997f2a5b9b1g-254.jpg?height=277&width=523&top_left_y=932&top_left_x=1365)

## 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_faa9db06d997f2a5b9b1g-254.jpg?height=349&width=545&top_left_y=1430&top_left_x=1359)