If the radii of two circles be 8 cm and 4 cm and the length of the transverse common tangent be 13 cm, then find the distance between the two centers is

Options

A. `sqrt(313)cm`
B. `sqrt(125)` cm
C. `5sqrt2` cm
D. `sqrt(135)` cm

Answer

To find the distance between the centers of two circles given their radii and the length of the transverse common tangent, we can use the following formula:

\[ AB = \sqrt{OP^2 - (R_1 + R_2)^2} \]

Where:
- \( AB \) is the length of the transverse common tangent.
- \( OP \) is the distance between the centers of the 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, \( R_2 = 4 \) cm
- Length of the transverse common tangent, \( AB = 13 \) cm

2. **Calculate the sum of the radii:**
\[ R_1 + R_2 = 8 + 4 = 12 \text{ cm} \]

3. **Substitute the values into the formula:**
\[ 13 = \sqrt{OP^2 - (12)^2} \]

4. **Square both sides to eliminate the square root:**
\[ 13^2 = OP^2 - 12^2 \]
\[ 169 = OP^2 - 144 \]

5. **Rearrange the equation to solve for \( OP^2 \):**
\[ OP^2 = 169 + 144 \]
\[ OP^2 = 313 \]

6. **Take the square root to find \( OP \):**
\[ OP = \sqrt{313} \text{ cm} \]

### Final Answer:
The distance between the two centers is \( \sqrt{313} \) cm.

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