8. Quadrilateral $A B C D$ has right angles at $A$ and $D$. Inside the quadrilateral, there is a circle with radius 10 that touches all sides of the quadrilateral. Side $B C$ has a length of 24. Point $E$ is the midpoint of $A D$ and point $F$ is the midpoint of $B C$.

What is the length of $E F$?
A) $\frac{43}{2}$
B) $\frac{13}{2} \sqrt{11}$
C) $\frac{33}{5} \sqrt{11}$
D) 22
E) $\frac{45}{2}$

![](https://cdn.mathpix.com/cropped/2024_04_17_f11aeda05102501e0c70g-2.jpg?height=400&width=483&top_left_y=2176&top_left_x=1523)

## B-questions
A8. D) 22 The points of tangency of $C D, A B$ and $B C$ to the circle we call $G, H$ and $J$. The center of the circle we call $O$. The length of $C G$ is $x$ and the length of $B H$ is $y$.

Since quadrilateral $O J C G$ is symmetrical (in $O C$), we see that $|C J|=x$. Similarly, $O H B J$ is symmetrical and we find $|B J|=y$. Therefore, we see that $x+y=24$.

The length of $A B$ is equal to $|A H|+|B H|=10+y$ and the length

![](https://cdn.mathpix.com/cropped/2024_04_17_732d64300714d2dca006g-2.jpg?height=371&width=457&top_left_y=1942&top_left_x=1371)
of $C D$ is equal to $|C G|+|D G|=10+x$. Since $E F$ is exactly halfway in the trapezoid $A B C D$, the length of $E F$ is exactly the average of the length of $A B$ and the length of $C D$. Therefore, we see that $|E F|=10+\frac{1}{2}(x+y)=22$.