Problem 11.4. Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two pieces so that he could form the outline of a right-angled triangle with the two pieces of this stick and the two whole sticks. How many square centimeters is the area of this triangle?
Answer: 216.

Solution. Let Senya break one of the sticks into parts of lengths $a$ and $24-a$, which we will call small, while sticks of length 24 will be called large. From four sticks, Senya formed a triangle, so one of the sides consists of two sticks, and the other two sides consist of one stick each.

Let's see which two sticks can be put together. If we put together two small sticks, we get an equilateral triangle, but we need a right triangle. If we put together two large sticks into one side, it will be greater than the sum of the other two sides (which contradicts the triangle inequality). The only remaining option is to combine a large and a small stick, and the other two sides will be just a small stick and a large stick (and these will be the legs, as they are shorter than the first side).

![](https://cdn.mathpix.com/cropped/2024_05_06_6da73bfd3e09e8b55e3fg-46.jpg?height=389&width=493&top_left_y=1255&top_left_x=480)

Fig. 9: to the solution of problem 11.4

Thus, the lengths of the sides of the right triangle are $a, 24$, and $48-a$ (Fig. 9). From the Pythagorean theorem, we get $a^{2}+24^{2}=(48-a)^{2}$. Transforming this equation, we find that $a=18$. Therefore, the area of the triangle is $\frac{18 \cdot 24}{2}=216$.