3. The villages of Arkadino, Borisovo, and Vadimovo are connected by straight roads. A square field adjoins the road between Arkadino and Borisovo, one side of which completely coincides with this road. A rectangular field adjoins the road between Borisovo and Vadimovo, one side of which completely coincides with this road, and the other side is 4 times longer. A rectangular forest adjoins the road between Arkadino and Vadimovo, one side of which completely coincides with this road, and the other side is 12 km. The area of the forest is 45 sq. km greater than the sum of the areas of the fields. Find the total area of the forest and the fields in sq. km.

$\{135\}$
Solution. The condition of the problem can be expressed by the following relation:

$r^{2}+4 p^{2}+45=12 q$

where $p, q, r$ are the lengths of the roads opposite the settlements Arkadino, Borisovo, and Vadimovo, respectively. This condition is in contradiction with the triangle inequality:

$r+p>q \Rightarrow 12 r+12 p>12 q \Rightarrow 12 r+12 p>r^{2}+4 p^{2}+45 \Rightarrow(r-6)^{2}+(2 p-3)^{2}<0$.

From this, it follows that all three settlements lie on the same straight line.

Moreover, $r=6, p=1.5, q=7.5$.

The total area is the sum: $r^{2}+4 p^{2}+12 q=36+9+90=135$.

Answer: 135