Two poles stand on the ground, which their feet `24` m apart. If these poles are respectively `12` m abd `19` m in height, find the distance between their tops.

Answer

To find the distance between the tops of the two poles, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two poles, one of height 12 m (let's call it pole AB) and the other of height 19 m (let's call it pole CD). The distance between the feet of the two poles (points A and C) is 24 m.

2. **Draw a Diagram**: Visualize the situation by drawing a diagram. Place point A at the base of pole AB, point B at the top of pole AB (12 m above A), point C at the base of pole CD, and point D at the top of pole CD (19 m above C). The distance AC is 24 m.

3. **Identify the Heights**: The height of pole AB (from A to B) is 12 m, and the height of pole CD (from C to D) is 19 m.

4. **Calculate the Vertical Distance Between the Tops of the Poles**: The vertical distance between points B and D can be calculated as:
\[ BD = CD - AB = 19 \, \text{m} - 12 \, \text{m} = 7 \, \text{m} \]

5. **Set Up the Right Triangle**: Now, we can form a right triangle BFD, where:
- BF (the vertical distance between the tops of the poles) = 7 m
- AC (the horizontal distance between the feet of the poles) = 24 m

6. **Apply the Pythagorean Theorem**: To find the distance BD (the hypotenuse of triangle BFD), we use the Pythagorean theorem:
\[ BD^2 = BF^2 + AC^2 \]
Substituting the values:
\[ BD^2 = 7^2 + 24^2 \]
\[ BD^2 = 49 + 576 \]
\[ BD^2 = 625 \]

7. **Calculate BD**: Now, take the square root of both sides to find BD:
\[ BD = \sqrt{625} = 25 \, \text{m} \]

### Final Answer:
The distance between the tops of the two poles (BD) is **25 meters**.