Two circles having radii `r_1` and `r_2` passing through vertex A of triangle ABC. One of the circle touches the side BC at B and the other circle touches the side BC at C. If a =5cm and `A=30^@` find `sqrt(r_1r_2)`

Answer

To solve the problem, we need to find \( \sqrt{r_1 r_2} \) given that two circles pass through vertex A of triangle ABC, one touches side BC at B and the other touches side BC at C. We are given that \( a = 5 \) cm and \( A = 30^\circ \).

### Step-by-Step Solution:

1. **Understanding the Triangle and Circles**:
- We have triangle ABC with \( A = 30^\circ \) and side \( a = BC = 5 \) cm.
- Let the two circles be \( C_1 \) and \( C_2 \) with radii \( r_1 \) and \( r_2 \) respectively.

2. **Using the Sine Rule**:
- According to the sine rule, we have:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
- Here, \( a = 5 \) cm and \( A = 30^\circ \), so:
\[ \frac{5}{\sin 30^\circ} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
- Since \( \sin 30^\circ = \frac{1}{2} \):
\[ \frac{5}{\frac{1}{2}} = 10 \implies a = 10 \]

3. **Finding \( r_1 \) and \( r_2 \)**:
- For circle \( C_1 \) (touching at B):
\[ r_1 = \frac{a \cdot \sin B}{2 \cdot \sin A} \]
- For circle \( C_2 \) (touching at C):
\[ r_2 = \frac{a \cdot \sin C}{2 \cdot \sin A} \]

4. **Expressing \( r_1 \) and \( r_2 \)**:
- From the sine rule, we have:
\[ b = 10 \sin B, \quad c = 10 \sin C \]
- Thus:
\[ r_1 = \frac{5 \cdot \sin B}{\sin 30^\circ} = 10 \sin B \]
\[ r_2 = \frac{5 \cdot \sin C}{\sin 30^\circ} = 10 \sin C \]

5. **Finding \( \sqrt{r_1 r_2} \)**:
- Now we can find \( \sqrt{r_1 r_2} \):
\[ \sqrt{r_1 r_2} = \sqrt{(10 \sin B)(10 \sin C)} = 10 \sqrt{\sin B \sin C} \]

6. **Using the Angle Sum Identity**:
- Since \( A + B + C = 180^\circ \), we have \( B + C = 150^\circ \).
- Using the identity \( \sin B \sin C = \frac{1}{2} (\cos(B - C) - \cos(B + C)) \):
- We can find \( \sin B \sin C \) using \( \sin(150^\circ) = \frac{1}{2} \).

7. **Final Calculation**:
- Thus:
\[ \sqrt{r_1 r_2} = 10 \cdot \sqrt{\frac{1}{2}} = 10 \cdot \frac{1}{\sqrt{2}} = 5\sqrt{2} \]

### Conclusion:
The value of \( \sqrt{r_1 r_2} \) is \( 5\sqrt{2} \) cm.