Math Problem Statement

Identify five segments skew to BC using the figure shown. Assume lines and planes that appear to be parallel or perpendicular are parallel or perpendicular, respectively.

Solution

The problem asks to identify five segments skew to line segment BC\overline{BC}. Skew lines are lines that are neither parallel nor intersecting, and they lie in different planes.

Step-by-step analysis:

  1. Identify BC\overline{BC}: This segment is in the top plane of the rectangular prism, connecting points BB and CC.

  2. Parallel lines: Lines that are parallel to BC\overline{BC} must be in the same direction and plane (the top face of the prism). This includes lines like EF\overline{EF} (on the bottom face) or other lines running in the same orientation.

  3. Perpendicular lines: Lines that are perpendicular to BC\overline{BC} either go straight vertically downwards (e.g., BF\overline{BF}) or horizontally across in a way that forms a 90-degree angle with BC\overline{BC}.

  4. Skew lines: These are lines that do not lie in the same plane as BC\overline{BC} and do not intersect it. These could be in the bottom face or connecting verticals that do not align with BC\overline{BC}.

Now let's check the options:

  • Option A: AE,DH,HF,EF,DC\overline{AE}, \overline{DH}, \overline{HF}, \overline{EF}, \overline{DC}

    • DC\overline{DC} is not skew since it is adjacent to BC\overline{BC} (they meet at point CC).
  • Option B: AE,EH,HG,EF,DH\overline{AE}, \overline{EH}, \overline{HG}, \overline{EF}, \overline{DH}

    • All these segments lie in different planes from BC\overline{BC} and do not intersect or run parallel to it, making them all skew.
  • Option C: AF,EC,HG,BF,DH\overline{AF}, \overline{EC}, \overline{HG}, \overline{BF}, \overline{DH}

    • BF\overline{BF} is not skew, as it is perpendicular to BC\overline{BC} and lies in the same plane.
  • Option D: BE,EF,HC,AD,DH\overline{BE}, \overline{EF}, \overline{HC}, \overline{AD}, \overline{DH}

    • BE\overline{BE} is not skew, as it shares point BB with BC\overline{BC}.

Correct answer:

  • Option B: AE,EH,HG,EF,DH\overline{AE}, \overline{EH}, \overline{HG}, \overline{EF}, \overline{DH}

All these segments are skew to BC\overline{BC} because they neither intersect nor are parallel to BC\overline{BC}, and they lie in different planes.

Would you like more details on skew lines or clarification on the solution? Here are five related questions for further exploration:

  1. What are skew lines, and how are they different from parallel and intersecting lines?
  2. How can we determine if two lines are parallel in a three-dimensional figure?
  3. Can two skew lines ever become parallel by rotating the figure?
  4. How do perpendicular lines differ from skew lines in terms of planes?
  5. Are there real-world examples where skew lines are commonly found, like in architecture?

Tip: When working with 3D figures, it helps to visualize or draw each face and identify the planes where lines exist to better understand the spatial relationships.

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Math Problem Analysis

Mathematical Concepts

Geometry
3D Figures
Skew Lines
Parallel and Perpendicular Lines

Formulas

-

Theorems

Skew Line Theorem

Suitable Grade Level

Grades 9-11