6. At Matthijs' table tennis club, they always place their ping pong balls on a table with cylindrical ball holders. Below is a side view of a ping pong ball on top of a ball holder. The bottom of the ball touches the table exactly. It is known that the ball holder is 4 centimeters wide and 1 centimeter high.

How many centimeters is the radius of the ball? Note: the picture is not to scale.
A) $2 \frac{1}{3}$
B) $2 \frac{1}{2}$
C) $2 \frac{2}{3}$
D) $2 \frac{5}{6}$
E) 3

![](https://cdn.mathpix.com/cropped/2024_04_17_56b78c3c74539692475ag-2.jpg?height=400&width=377&top_left_y=1159&top_left_x=1528)
A6. B) $2 \frac{1}{2}$ We draw an auxiliary line from the center of the ball to the edge of the holder, and from the center of the ball to the table. We also draw a line parallel to the table. We are looking for $s$, the radius of the ball.

We see a right triangle with legs 2 and $s-1$ and hypotenuse $s$. The Pythagorean theorem gives that $2^{2}+(s-1)^{2}=s^{2}$. Simplifying gives $4+s^{2}-2 s+1=s^{2}$, or $2 s=5$. Thus, $s=\frac{5}{2}=2 \frac{1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_a4374d618d1fd957f1eeg-3.jpg?height=397&width=377&top_left_y=196&top_left_x=1528)