Two poles of height 9 m and 15 m stand vertically upright on a plane ground. If the distance between their tops is 10m, then find the distance between their feet.

Answer

To solve the problem of finding the distance between the feet of two poles of heights 9 m and 15 m, given that the distance between their tops is 10 m, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- Let the height of the first pole (A) be 9 m and the height of the second pole (B) be 15 m.
- The distance between the tops of the poles (C) is given as 10 m.

2. **Draw a Diagram**:
- Draw two vertical lines representing the poles.
- Mark the top of the first pole as point A (9 m) and the top of the second pole as point B (15 m).
- The horizontal distance between the feet of the poles will be denoted as DE, where D is the foot of the first pole and E is the foot of the second pole.

3. **Identify the Right Triangle**:
- The points D, E, and C form a right triangle, where:
- AC (the vertical distance from A to C) = 9 m (height of the first pole),
- BC (the vertical distance from B to C) = 15 m (height of the second pole),
- The distance between the tops of the poles (AB) = 10 m.

4. **Calculate the Vertical Difference**:
- The vertical difference between the two poles is:
\[ BC - AC = 15 m - 9 m = 6 m \]
- This means that the vertical leg of the triangle (BC) is 6 m.

5. **Apply the Pythagorean Theorem**:
- In triangle ABC, we can apply the Pythagorean theorem:
\[ AB^2 = AC^2 + BC^2 \]
- Substituting the known values:
\[ 10^2 = x^2 + 6^2 \]
\[ 100 = x^2 + 36 \]

6. **Solve for x**:
- Rearranging the equation gives:
\[ x^2 = 100 - 36 \]
\[ x^2 = 64 \]
\[ x = \sqrt{64} = 8 \]

7. **Conclusion**:
- Therefore, the distance between the feet of the two poles (DE) is 8 m.

### Final Answer:
The distance between the feet of the two poles is **8 meters**.