The problem is written in Ukrainian and can be translated as follows:

"In a right-angled triangle, the point where the incircle touches the hypotenuse divides the hypotenuse into segments of lengths 5 cm and 12 cm. Find the legs of the triangle."

Solution:

Let the legs of the right triangle be $a$ and $b$, and the hypotenuse be $c$.

The point where the incircle touches the hypotenuse divides it into two segments, $p$ and $q$, where $p = 5$ cm and $q = 12$ cm.

We know the following relationships:

1. $p + q = c$ (since $p$ and $q$ are segments of the hypotenuse)
2. $a = s - p$ and $b = s - q$, where $s = \frac{a + b + c}{2}$ is the semiperimeter.

From the first relationship, we have: $c = 5 + 12 = 17 \text{ cm}$

Now, the semiperimeter $s$ is given by: $s = \frac{a + b + c}{2}$

Substituting the expressions for $a$ and $b$ in terms of $s$, $p$, and $q$: $a = s - p = s - 5$ $b = s - q = s - 12$

Since $s = \frac{a + b + c}{2}$, we can write: $a + b = 2s - c = 2s - 17$

Substitute the expressions for $a$ and $b$ into the Pythagorean theorem: $a^2 + b^2 = c^2$

Substituting the values: $(17 - 5)^2 + (17 - 12)^2 = 17^2$ $12^2 + 5^2 = 289$ $144 + 25 = 169$

Thus, the correct lengths of the legs $a$ and $b$ are 12 cm and 5 cm respectively.

Would you like more details, or do you have any questions?

Here are 5 related questions:

1. What is the formula for finding the radius of the incircle of a right triangle?
2. How can you find the area of a right triangle using the legs and hypotenuse?
3. What are the properties of a 30-60-90 triangle?
4. How would you solve this problem if only the hypotenuse was given?
5. What is the relationship between the semiperimeter and the legs of a right triangle?

Tip: In right-angled triangles, the incircle touches the hypotenuse at a point that divides it into segments equal to the differences between the semiperimeter and the other sides.