The problem involves using the concept of similar triangles, which is related to the theorem of Thales. According to the image, there are two right triangles formed by the shadows of the statue and the pole.

Given Data:

1. Height of the statue: 4.2 m
2. Height of the pole: 5.8 m
3. Length of the shadow of the statue: 3.4 m
4. Length of the shadow of the pole: $x$ (unknown)

Step-by-Step Solution:

We use the theorem of Thales which states that corresponding sides of similar triangles are proportional.

From the two triangles, we can set up the proportion:

$\frac{4.2}{3.4} = \frac{5.8}{x}$

Now, let's solve for $x$:

$x = \frac{5.8 \times 3.4}{4.2}$

Calculating this value:

$x \approx 4.7 \, \text{m}$

Therefore, the distance between the pole and the statue is approximately $x - 3.4 = 4.7 - 3.4 = 1.3 \, \text{m}$.

Conclusion:

The approximate distance between the pole and the statue is 1.3 m.

Would you like further details or explanations on this solution?

Related Questions:

1. How can we apply Thales' theorem in other geometric problems?
2. What are the real-world applications of similar triangles?
3. How do we find unknown lengths in similar figures?
4. How does the concept of proportionality apply to scale models?
5. What are other geometric theorems related to similar triangles?

Tip: When solving problems with similar triangles, always look for corresponding sides and angles to set up proportional relationships!