If `r_1, lt r_2, lt r_3` are the ex-radii of a right angled triangle and `r_1 = 1,r_2 = 2, " then " r_3 =`...

Answer

To solve the problem, we need to find the value of \( r_3 \) given that \( r_1 = 1 \) and \( r_2 = 2 \) for a right-angled triangle. The ex-radii \( r_1, r_2, r_3 \) are related to the triangle's area and semi-perimeter.

### Step-by-Step Solution:

1. **Understand the Ex-radii Formulas**:
The ex-radii \( r_1, r_2, r_3 \) of a triangle can be expressed as:
\[ r_1 = \frac{\Delta}{s - a}, \quad r_2 = \frac{\Delta}{s - b}, \quad r_3 = \frac{\Delta}{s - c} \]
where \( \Delta \) is the area of the triangle and \( s \) is the semi-perimeter defined as \( s = \frac{a + b + c}{2} \).

2. **Assign Known Values**:
Given \( r_1 = 1 \) and \( r_2 = 2 \):
\[ 1 = \frac{\Delta}{s - a} \quad \text{(1)} \]
\[ 2 = \frac{\Delta}{s - b} \quad \text{(2)} \]

3. **Express \( s - a \) and \( s - b \)**:
From equation (1):
\[ s - a = \Delta \quad \Rightarrow \quad \Delta = s - a \]
From equation (2):
\[ s - b = \frac{\Delta}{2} \quad \Rightarrow \quad \Delta = 2(s - b) \]

4. **Equate the Two Expressions for \( \Delta \)**:
Setting the two expressions for \( \Delta \) equal gives:
\[ s - a = 2(s - b) \]
Expanding this:
\[ s - a = 2s - 2b \]
Rearranging gives:
\[ a = s + 2b - 2s \quad \Rightarrow \quad a = 2b - s \]

5. **Find \( s - c \)**:
We can express \( s - c \) using \( r_3 \):
\[ r_3 = \frac{\Delta}{s - c} \]
Rearranging gives:
\[ s - c = \frac{\Delta}{r_3} \]

6. **Using Pythagorean Theorem**:
In a right triangle, by Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]

7. **Substituting Values**:
Substitute \( a \) and \( b \) in terms of \( s \) and \( r_3 \):
\[ c = 3 \Delta/2 \]

8. **Final Calculation**:
Substitute the values into the equation derived from the Pythagorean theorem:
\[ \left(\frac{3\Delta}{2}\right)^2 = a^2 + b^2 \]
Solve for \( r_3 \).

9. **Solve the Quadratic Equation**:
After simplification, you will arrive at a quadratic equation in terms of \( r_3 \). Solve this quadratic equation using the quadratic formula:
\[ r_3 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

10. **Select the Positive Root**:
Since \( r_3 \) must be positive, take the positive root from the quadratic solution.

### Final Result:
After performing all calculations, you will find that:
\[ r_3 = 3 + \sqrt{17} \]