Math Problem Statement

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

Solution

The question in the image is in Ukrainian and reads:

"Пряма mm лежить у площині α\alpha і перетинає площину β\beta. Яке взаємне розташування площин α\alpha і β\beta?"

Translated to English:

"A line mm lies in the plane α\alpha and intersects plane β\beta. What is the mutual arrangement of the planes α\alpha and β\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 mm lies in plane α\alpha and intersects plane β\beta, the two planes α\alpha and β\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).


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.

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Math Problem Analysis

Mathematical Concepts

Plane Geometry
3D Geometry

Formulas

Equations of Planes in 3D Space

Theorems

Intersection of Planes

Suitable Grade Level

Grades 10-12