Two circles of radil 5 cm and 3cm touch externally, then the ratio in which the direct common tangent to the circles dtvides externally the line joining the centers of the circles is:

Options

A. `5:3`
B. `3:5`
C. `2.5:1.5`
D. `1.5 : 2.5 `

Answer

To solve the problem of finding the ratio in which the direct common tangent to two externally touching circles divides the line joining their centers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The radius of the first circle (R1) = 5 cm
- The radius of the second circle (R2) = 3 cm

2. **Understand the Configuration:**
- Let the centers of the two circles be A and B.
- The distance between the centers A and B can be calculated as the sum of the radii since the circles touch externally:
\[ AB = R1 + R2 = 5 \, \text{cm} + 3 \, \text{cm} = 8 \, \text{cm} \]

3. **Identify the Points:**
- Let P be the point where the direct common tangent touches the line joining the centers A and B.
- We need to find the ratio in which the tangent divides the line segment AB.

4. **Use Similar Triangles:**
- The triangles formed by the tangent and the radii to the points of tangency are similar.
- Specifically, triangles PBD and PAC are similar, where D is the point where the tangent touches the first circle and C is the point where the tangent touches the second circle.

5. **Set Up the Ratio Using Similar Triangles:**
- From the properties of similar triangles, we have:
\[ \frac{PB}{PA} = \frac{BD}{AC} \]
- Here, BD = R2 = 3 cm and AC = R1 = 5 cm.

6. **Substituting Values:**
- Substitute the known values into the ratio:
\[ \frac{PB}{PA} = \frac{3}{5} \]

7. **Finding the Ratio of AP to BP:**
- Since we need the ratio of AP to BP, we can express it as:
\[ \frac{AP}{BP} = \frac{PB}{PA}^{-1} = \frac{5}{3} \]

8. **Final Result:**
- Therefore, the ratio in which the direct common tangent divides the line joining the centers of the circles externally is:
\[ AP : BP = 5 : 3 \]