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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.

Learn how to solve a right triangle problem involving the incircle and segments of the hypotenuse. This problem explores key geometric concepts including the Pythagorean theorem and the properties of right triangles. Suitable for high school students.

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