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 $A F=9, B F=6$.
# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_a3feddd7c418bd26b633g-3.jpg?height=346&width=1094&top_left_y=521&top_left_x=524)

Let the lines $A E$ and $B C$ intersect at point $K$, then triangles $A E D$ and $K C E$ are congruent ( $\angle A E D = \angle C E D$ 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 $A K$ ), hence $A F = F K = 7$.

Since $A D = C K$ and $A D = B C$, we get $B K = B F + F K = 2(B F + F C)$. $F C = 7.5 - 6 = 1.5 \text{ cm}$.

Answer: 1.5 cm.