The centres of two circles with radii 6 cm and 2 cm are 10 cm apart. Calculate the length of the transverse common tangent.

Options

A. 5cm
B. 6cm
C. 7cm
D. 2cm

Answer

To find the length of the transverse common tangent between two circles with given radii and distance between their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the first circle (R1) = 6 cm
- Radius of the second circle (R2) = 2 cm
- Distance between the centers of the circles (d) = 10 cm

2. **Understand the Formula for the Length of the Transverse Common Tangent:**
The formula for the length of the transverse common tangent (TQ) between two circles is given by:
\[ TQ = \sqrt{d^2 - (R1 + R2)^2} \]

3. **Calculate the Sum of the Radii:**
Calculate \( R1 + R2 \):
\[ R1 + R2 = 6 + 2 = 8 \text{ cm} \]

4. **Square the Distance and the Sum of the Radii:**
- Square the distance between the centers:
\[ d^2 = 10^2 = 100 \]
- Square the sum of the radii:
\[ (R1 + R2)^2 = 8^2 = 64 \]

5. **Substitute into the Formula:**
Now substitute the values into the formula:
\[ TQ = \sqrt{100 - 64} \]

6. **Calculate the Result:**
Simplify the expression:
\[ TQ = \sqrt{36} = 6 \text{ cm} \]

7. **Final Answer:**
Therefore, the length of the transverse common tangent is:
\[ \text{Length of transverse common tangent} = 6 \text{ cm} \]