A circumcircle is a circle which passes through all vertices of a triangle and an incircle is a circle which is inscribed in a triangle touching all sides of a triangle. Let ABC be a right angled triangle whose radius of circumcircle is 5 and its one side AB = 6. The radius of incircle of triangle ABC is r. The value of r is

Answer

To solve the problem regarding the circumcircle and incircle of a right-angled triangle ABC, we will follow these steps:

### Step 1: Understand the properties of the triangle
Given that triangle ABC is a right-angled triangle, we know that the circumradius \( R \) can be calculated using the formula:
\[ R = \frac{c}{2} \]
where \( c \) is the hypotenuse of the triangle.

### Step 2: Use the circumradius to find the hypotenuse
We are given that the circumradius \( R = 5 \). Therefore, we can set up the equation:
\[ 5 = \frac{c}{2} \]
Multiplying both sides by 2 gives us:
\[ c = 10 \]

### Step 3: Use the Pythagorean theorem
In a right-angled triangle, the relationship between the sides can be expressed using the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Let \( AB = 6 \) (one side) and \( AC = c = 10 \) (the hypotenuse). We need to find the length of the other side \( BC \). Substituting the known values, we have:
\[ 6^2 + BC^2 = 10^2 \]
Calculating the squares:
\[ 36 + BC^2 = 100 \]
Subtracting 36 from both sides gives:
\[ BC^2 = 64 \]
Taking the square root:
\[ BC = 8 \]

### Step 4: Calculate the radius of the incircle
The radius \( r \) of the incircle of a triangle can be calculated using the formula:
\[ r = \frac{A}{s} \]
where \( A \) is the area of the triangle and \( s \) is the semi-perimeter.

### Step 5: Calculate the semi-perimeter \( s \)
The semi-perimeter \( s \) is given by:
\[ s = \frac{AB + BC + AC}{2} = \frac{6 + 8 + 10}{2} = \frac{24}{2} = 12 \]

### Step 6: Calculate the area \( A \)
The area \( A \) of triangle ABC can be calculated as:
\[ A = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 6 \times 8 = 24 \]

### Step 7: Substitute values to find \( r \)
Now substituting the values of \( A \) and \( s \) into the formula for \( r \):
\[ r = \frac{A}{s} = \frac{24}{12} = 2 \]

### Conclusion
Thus, the radius of the incircle \( r \) is:
\[ \boxed{2} \]