Khachaturyan A.V.

A rectangular sheet of paper was folded, aligning a vertex with the midpoint of the opposite shorter side (see figure). It turned out that triangles I and II are equal. Find the longer side of the rectangle if the shorter side is 8.

![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-08.jpg?height=586&width=552&top_left_y=576&top_left_x=753)

#
Let's mark the equal segments (see fig. - here we used the fact that in congruent triangles, sides opposite equal angles are equal).

![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-08.jpg?height=723&width=603&top_left_y=1376&top_left_x=722)

We see that the length of the smaller side is $a+b$. Therefore, $a+b=8$ and the larger side has a length of $a+b+4=$ $8+4=12$.

## Answer

12.