https://file.aiquickdraw.com/mathbot/file/1726742034874o2o7c43a.jpg

The question in the image is in Ukrainian and reads:

"Пряма \( m \) лежить у площині \( \alpha \) і перетинає площину \( \beta \). Яке взаємне розташування площин \( \alpha \) і \( \beta \)?"

Translated to English:

"A line \( m \) lies in the plane \( \alpha \) and intersects plane \( \beta \). What is the mutual arrangement of the planes \( \alpha \) and \( \beta \)?"

The answer choices are:
1. Площини не перетинаються (The planes do not intersect)
2. Визначити неможливо (It is impossible to determine)
3. Перетинаються по деякій прямій (They intersect along a line)
4. Мають одну спільну точку (They have one common point)
5. Площини збігаються (The planes coincide)

### Explanation:
- Since line \( m \) lies in plane \( \alpha \) and intersects plane \( \beta \), the two planes \( \alpha \) and \( \beta \) must intersect. Typically, when two planes intersect, they do so along a line.
  
Thus, the correct answer is: **Перетинаються по деякій прямій (They intersect along a line)**.

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Would you like further details or clarifications?

Here are 5 related questions to deepen understanding:
1. What are the conditions for two planes to intersect along a line?
2. What happens geometrically when two planes have one common point?
3. How do we determine if two planes coincide in three-dimensional space?
4. How can we mathematically express the intersection of two planes using their equations?
5. How can a line being in one plane and intersecting another affect the mutual arrangement of the planes?

**Tip:** In 3D geometry, two distinct planes that intersect do so along a line unless they are parallel or coincide.

Пряма m лежить у площині α і перетинає площину β. Яке взаємне розташування площин α і β?

Explore the relationship between two planes when a line lies in one plane and intersects another. This problem covers 3D geometry concepts, including the intersection of planes and conditions for planes to intersect along a line. Suitable for students in Grades 10-12.

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