Problem 4. A rectangular sheet of paper was folded, aligning a vertex with the midpoint of the opposite shorter side (Fig. 12). It turned out that triangles I and II are equal. Find the longer side of the rectangle if the shorter side is 8.

[6 points] (A. V. Khachatryan)
Answer: 12.

![](https://cdn.mathpix.com/cropped/2024_05_06_07f23e5c5625e3aacdd0g-3.jpg?height=458&width=437&top_left_y=1316&top_left_x=871)

Fig. 12

Solution. Let's mark the equal segments (Fig. 13 - here we used the fact that in congruent triangles, sides opposite equal angles are equal). We see that the length of the larger side is $a+b+4$, and the length of the smaller side is $a+b$. Therefore, $a+b=8$ and the larger side has a length of $a+b+4=8+4=12$.

![](https://cdn.mathpix.com/cropped/2024_05_06_07f23e5c5625e3aacdd0g-4.jpg?height=523&width=437&top_left_y=352&top_left_x=530)

Fig. 13

Comment. By applying the Pythagorean theorem, we can find the lengths of the sides of triangles I and II. It turns out that these are Egyptian triangles - triangles with sides 3, 4, and 5.