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Let's solve the problem step by step.

The problem is in Persian and it reads:
"In a right triangle with sides of lengths 3, 4, and 5 units, what is the distance from the center of the hypotenuse to the midpoint of the hypotenuse?"

We have a right triangle with legs of lengths 3 and 4 units, and a hypotenuse of length 5 units. This is a well-known Pythagorean triple. Let's label the triangle:
- The vertices are \( A \), \( B \), and \( C \).
- \( AB = 3 \), \( BC = 4 \), and \( AC = 5 \).

The center of the hypotenuse, \( M \), is the midpoint of the hypotenuse \( AC \). The coordinates of the vertices can be assumed as:
- \( A(0, 0) \)
- \( B(3, 0) \)
- \( C(0, 4) \)

The midpoint \( M \) of hypotenuse \( AC \) can be calculated as:
\[ M \left( \frac{0+3}{2}, \frac{0+4}{2} \right) = \left( \frac{3}{2}, 2 \right) \]

Now, we need to calculate the distance from \( M \left( \frac{3}{2}, 2 \right) \) to the midpoint of the hypotenuse, which is point \( C(0, 4) \).

We use the distance formula:
\[ d = \sqrt{ \left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 } \]

Let's substitute the points \( \left( \frac{3}{2}, 2 \right) \) and \( (0, 4) \):
\[ d = \sqrt{ \left( 0 - \frac{3}{2} \right)^2 + \left( 4 - 2 \right)^2 } \]
\[ d = \sqrt{ \left( -\frac{3}{2} \right)^2 + 2^2 } \]
\[ d = \sqrt{ \frac{9}{4} + 4 } \]
\[ d = \sqrt{ \frac{9}{4} + \frac{16}{4} } \]
\[ d = \sqrt{ \frac{25}{4} } \]
\[ d = \frac{5}{2} \]

The correct answer is \( \frac{5}{2} \), which matches one of the given choices (option 2).

Do you want details or have any questions? Here are some questions you might consider:

1. What is the midpoint formula for a line segment?
2. How do you apply the distance formula between two points?
3. Can you explain the Pythagorean theorem?
4. What are some other common Pythagorean triples?
5. How can you find the centroid of a triangle?

**Tip:** Understanding the basic properties of right triangles and common Pythagorean triples can greatly simplify solving many geometry problems.

Learn how to calculate the distance from the center of the hypotenuse to the midpoint in a right triangle with sides 3, 4, and 5 units. This problem involves applying the Pythagorean theorem, midpoint formula, and distance formula. Suitable for high school students.

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