If the length of the transverse tangent is 8 cm, then calculate the centre distance (in cm) of the two circles with radii 7 cm and 3 cm.

Options

A. `sqrt(100)`
B. `sqrt(120)`
C. `sqrt(164)`
D. `sqrt(180)`

Answer

To solve the problem, we need to find the center distance \( d \) between two circles given the length of the transverse tangent and the radii of the circles.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of the transverse tangent \( PQ = 8 \) cm
- Radius of the first circle \( R_1 = 7 \) cm
- Radius of the second circle \( R_2 = 3 \) cm

2. **Use the Formula for the Length of the Transverse Tangent:**
The formula for the length of the transverse tangent between two circles is given by:
\[ PQ^2 = d^2 - (R_1 + R_2)^2 \]
where \( d \) is the distance between the centers of the two circles.

3. **Substitute the Known Values into the Formula:**
Substitute \( PQ = 8 \) cm, \( R_1 = 7 \) cm, and \( R_2 = 3 \) cm into the formula:
\[ 8^2 = d^2 - (7 + 3)^2 \]

4. **Calculate the Squares:**
- Calculate \( 8^2 = 64 \)
- Calculate \( (7 + 3)^2 = 10^2 = 100 \)

5. **Set Up the Equation:**
Now, we can set up the equation:
\[ 64 = d^2 - 100 \]

6. **Solve for \( d^2 \):**
Rearranging the equation gives:
\[ d^2 = 64 + 100 \]
\[ d^2 = 164 \]

7. **Find \( d \):**
Taking the square root of both sides:
\[ d = \sqrt{164} \]

8. **Final Answer:**
Thus, the center distance \( d \) between the two circles is:
\[ d = \sqrt{164} \text{ cm} \]