7. Rectangle $A B C D$ is divided into squares. Side $A B$ has length 16.

![](https://cdn.mathpix.com/cropped/2024_04_17_01cb006263f00c7722c1g-2.jpg?height=503&width=574&top_left_y=1305&top_left_x=747)

How long is side $A D$?
A) 13
B) $\frac{27}{2}$
C) 14
D) $\frac{29}{2}$
E) 15
A7. D) $\frac{29}{2}$ In each of the small squares, we have indicated the side length with a letter. We see that the following holds: $2 a+2 d=16$ and thus $d=8-a$. In the same way, we can express the other side lengths in terms of $a$ using the following relationships that we see in the figure:

$$
\begin{aligned}
c & =2 a \\
b & =a+c-d=3 a-(8-a)=4 a-8, \\
e & =2 a+b=6 a-8, \\
f & =b+e=(4 a-8)+(6 a-8)=10 a-16 .
\end{aligned}
$$

Of course, it also holds that $e+f=16$, so we find that $(6 a-8)+(10 a-16)=16$. This gives $16 a=40$ and thus $a=\frac{5}{2}$.

The length of side $A D$ can now be expressed as

$$
|A D|=a+c+e=3 a+(6 a-8)=9 a-8=\frac{45}{2}-8=\frac{29}{2} \text {. }
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_fc21619166b069277c26g-2.jpg?height=506&width=571&top_left_y=2077&top_left_x=751)