7. In rectangle $A B C D$, point $E$ is the midpoint of side $C D$. On side $B C$, point $F$ is taken such that angle $A E F$ is a right angle. Find the length of segment $F C$, if $A F=7, B F=4$.
# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_de39ee82fbf77ee5c736g-3.jpg?height=268&width=813&top_left_y=323&top_left_x=670)

Let the lines $A E$ and $B C$ intersect at point $K$, then triangles $A E D$ and $K C E$ are equal ( $\angle A E D = \angle C E D$ as vertical angles, $C E = E D, \angle A D E = \angle K C E = 90^{\circ}$ ), therefore, $C K = A D, A E = E K$. Triangle $A F K$ is isosceles ( $F E \perp A K, E$ is the midpoint of $A K$ ), so $A F = F K = 7$. Since $A D = C K$ and $A D = B C$, we get $B K = B F + F K = 2(B F + F C) . F C = 5.5 - 4 = 1.5 \text{ cm}$. Answer: 1.5 cm.