If a right triangle has integer side lengths, is its inradius necessarily an integer?

Yes, if a right triangle has integer side lengths (i.e., it is a Pythagorean triangle), then its inradius is necessarily an integer.

### Explanation:

Consider a right triangle with integer side lengths \( a \), \( b \), and \( c \), where \( c \) is the hypotenuse. For such a triangle:

1. **Area \( A \):** The area is given by \( A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \).

2. **Semi-perimeter \( s \):** The semi-perimeter is \( s = \frac{a + b + c}{2} \).

3. **Inradius \( r \):** The inradius \( r \) is given by \( r = \frac{A}{s} \).

Given that \( a \), \( b \), and \( c \) are integers, the area \( A \) and the semi-perimeter \( s \) will also be integers. Since \( r \) is calculated as the ratio of two integers (the area \( A \) and the semi-perimeter \( s \)), \( r \) will be an integer, provided that the ratio simplifies to an integer.

### Proof by example:

- Consider the classic 3-4-5 triangle:
  - \( a = 3 \), \( b = 4 \), \( c = 5 \).
  - Area \( A = \frac{1}{2} \times 3 \times 4 = 6 \).
  - Semi-perimeter \( s = \frac{3 + 4 + 5}{2} = 6 \).
  - Inradius \( r = \frac{6}{6} = 1 \) (which is an integer).

- Consider the 5-12-13 triangle:
  - \( a = 5 \), \( b = 12 \), \( c = 13 \).
  - Area \( A = \frac{1}{2} \times 5 \times 12 = 30 \).
  - Semi-perimeter \( s = \frac{5 + 12 + 13}{2} = 15 \).
  - Inradius \( r = \frac{30}{15} = 2 \) (which is an integer).

Therefore, for any right triangle with integer side lengths, the inradius is indeed an integer.

Would you like further details or have any questions? Here are some related questions to consider:

1. How do you determine if a set of three integers can form a right triangle?
2. What are the properties of a Pythagorean triple?
3. Can the circumradius of a right triangle with integer sides also be an integer?
4. What is the relationship between the inradius and the semiperimeter in non-right triangles?
5. How can you generate Pythagorean triples using algebraic formulas?

**Tip:** The inradius of a right triangle with integer sides is always an integer, which can be useful in solving problems related to circle properties within triangles.

Understand whether the inradius of a right triangle with integer side lengths is necessarily an integer. Learn how to calculate the inradius using the area and semi-perimeter of the triangle, illustrated with examples like the classic 3-4-5 triangle and the 5-12-13 triangle.

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