In the figure, line $t$ is parallel to segment $E F$ and tangent to the circle. If $A E=12, A F=10$ and $F C=14$, determine the length of $E B$.

![](https://cdn.mathpix.com/cropped/2024_05_01_10107d2fba0ec26bc845g-12.jpg?height=662&width=811&top_left_y=520&top_left_x=520)
Solution

Since the line $t$ is tangent to the circle, we have $\angle X A E=\angle A C B$. Furthermore, since $t$ and $E F$ are parallel, we have $\angle X A E=\angle A E F$. Similarly, we have $\angle Y A F=\angle A F E=\angle A B C$. Therefore, $\triangle A F E \simeq \triangle A B C$. Thus,

$$
\begin{aligned}
\frac{A E}{A C} & =\frac{A F}{A B} \\
\frac{12}{10+14} & =\frac{10}{12+E B} \\
12+E B & =20 \\
E B & =8
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_10107d2fba0ec26bc845g-12.jpg?height=668&width=819&top_left_y=1899&top_left_x=516)