SR is dirrect common tangent of two circles whose radii are respectively 8 cm and 3 cm and centres are 13 cm apart. If S and R are points of contact, then the length of SR is

Options

A. 17 cm
B. 10 cm
C. 12 cm
D. 11 cm

Answer

To find the length of the direct common tangent \( SR \) between two circles with given radii and distance between their centers, we can use the formula for the length of the direct common tangent:

\[ L = \sqrt{d^2 - (r_1 - r_2)^2} \]

where:
- \( L \) is the length of the direct common tangent,
- \( d \) is the distance between the centers of the two circles,
- \( r_1 \) and \( r_2 \) are the radii of the two circles.

### Step-by-Step Solution:

1. **Identify the given values**:
- Radius of the first circle, \( r_1 = 8 \) cm
- Radius of the second circle, \( r_2 = 3 \) cm
- Distance between the centers, \( d = 13 \) cm

2. **Calculate the difference of the radii**:
\[ r_1 - r_2 = 8 - 3 = 5 \text{ cm} \]

3. **Square the distance between the centers**:
\[ d^2 = 13^2 = 169 \]

4. **Square the difference of the radii**:
\[ (r_1 - r_2)^2 = 5^2 = 25 \]

5. **Substitute these values into the formula**:
\[ L^2 = d^2 - (r_1 - r_2)^2 = 169 - 25 = 144 \]

6. **Take the square root to find the length of the tangent**:
\[ L = \sqrt{144} = 12 \text{ cm} \]

Thus, the length of the direct common tangent \( SR \) is **12 cm**.