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points, lines, and planes · practice example 1 refer to the figure for …

Question

points, lines, and planes · practice

example 1
refer to the figure for exercises 1-7.

  1. name the lines that are only in plane q.
  2. how many planes are labeled in the figure?
  3. name the plane containing the lines m and t.
  4. name the intersection of lines m and t.
  5. name a point that is not coplanar with points a, b, and c.
  6. are points f, m, g, and p coplanar? explain.
  7. does line n intersect line q? explain.

example 2
name the geometric terms modeled by each object or phrase.

  1. one solar panel
  2. a tabletop
  3. bridge support beam
  4. a chessboard
  5. image of a prism
  6. image of a line with a dot
  7. a wall and the floor
  8. the edge of a table
  9. two connected walls
  10. a blanket
  11. a telephone pole
  12. a computer screen

Explanation:

Response

Let's solve these problems one by one (we'll focus on a few key ones, like problem 2, 3, 4, 5, 8 - 19 which are about geometric terms and plane - line - point relationships):

Problem 2: How many planes are labeled in the figure?
Step 1: Identify labeled planes

Looking at the figure, the labeled planes are plane \(Q\) and plane \(R\) (and maybe we check again: the top plane is \(R\), the bottom one is \(Q\)? Wait, no, let's see the labels. Wait, the figure has plane \(R\) (with points \(A, B, C\) etc.) and plane \(Q\) (with points \(G, P, H\) etc.). Wait, maybe I missed? Wait, no, let's count: plane \(R\) and plane \(Q\)? Wait, no, maybe there are two? Wait, no, looking at the figure, the top plane is labeled \(R\), the bottom one is \(Q\), and is there another? Wait, the vertical line has points \(D, M, P\). Wait, maybe the answer is 2? Wait, no, let's re - examine. The figure shows two planes: \(R\) (the upper one) and \(Q\) (the lower one). So the number of labeled planes is 2? Wait, maybe I made a mistake. Wait, the problem says "labeled in the figure". Let's assume that the labeled planes are \(R\) and \(Q\), so the number is 2.

Problem 3: Name the plane containing the lines \(m\) and \(t\)
Step 1: Identify the plane with lines \(m\) and \(t\)

Lines \(m\) and \(t\) are on the upper plane. Looking at the labels, the upper plane is \(R\). So the plane containing lines \(m\) and \(t\) is plane \(R\).

Problem 4: Name the intersection of lines \(m\) and \(t\)
Step 1: Find the common point of lines \(m\) and \(t\)

Lines \(m\) and \(t\) intersect at point \(C\) (from the figure, where the lines cross). So the intersection of lines \(m\) and \(t\) is point \(C\).

Problem 5: Name a point that is not coplanar with points \(A, B,\) and \(C\)
Step 1: Recall coplanar definition

Coplanar points lie on the same plane. Points \(A, B, C\) are on plane \(R\). A point not on plane \(R\) would be non - coplanar with them. Looking at the figure, point \(M\) (or \(P\) or \(G\)): let's take point \(M\). Point \(M\) is on the vertical line and on plane \(Q\), not on plane \(R\) (where \(A, B, C\) are). So a point like \(M\) (or \(P\), \(G\)) is not coplanar with \(A, B, C\).

Problem 8: one solar panel
Step 1: Recall geometric terms

A solar panel is a flat, two - dimensional surface that extends infinitely (in the geometric model). So it models a plane.

Problem 9: a tabletop
Step 1: Recall geometric terms

A tabletop is a flat, two - dimensional surface. So it models a plane.

Problem 10: bridge support beam
Step 1: Recall geometric terms

A bridge support beam is a straight, one - dimensional object that extends infinitely (in the geometric model). So it models a line.

Problem 11: a chessboard
Step 1: Recall geometric terms

A chessboard is a flat, two - dimensional surface. So it models a plane.

Problem 14: a wall and the floor
Step 1: Recall geometric terms

A wall and the floor intersect in a line (the edge where they meet). So their intersection is a line.

Problem 15: the edge of a table
Step 1: Recall geometric terms

The edge of a table is a straight, one - dimensional object. So it models a line.

Problem 16: two connected walls
Step 1: Recall geometric terms

Two connected walls intersect in a line (the corner where they meet). So their intersection is a line.

Problem 17: a blanket
Step 1: Recall geometric terms

A blanket is a flat, two - dimensional surface. So it models a plane.

Problem 18: a telephone pole
Step 1: Recall g…

Answer:

s to specific problems:

  • Problem 2: The number of labeled planes is \(\boldsymbol{2}\) (planes \(R\) and \(Q\)).
  • Problem 3: The plane containing lines \(m\) and \(t\) is \(\boldsymbol{\text{Plane } R}\).
  • Problem 4: The intersection of lines \(m\) and \(t\) is \(\boldsymbol{\text{Point } C}\).
  • Problem 5: A point not coplanar with \(A, B, C\) is \(\boldsymbol{\text{Point } M}\) (or \(P\), \(G\)).
  • Problem 8: One solar panel models a \(\boldsymbol{\text{plane}}\).
  • Problem 9: A tabletop models a \(\boldsymbol{\text{plane}}\).
  • Problem 10: A bridge support beam models a \(\boldsymbol{\text{line}}\).
  • Problem 11: A chessboard models a \(\boldsymbol{\text{plane}}\).
  • Problem 14: A wall and the floor intersect in a \(\boldsymbol{\text{line}}\).
  • Problem 15: The edge of a table models a \(\boldsymbol{\text{line}}\).
  • Problem 16: Two connected walls intersect in a \(\boldsymbol{\text{line}}\).
  • Problem 17: A blanket models a \(\boldsymbol{\text{plane}}\).
  • Problem 18: A telephone pole models a \(\boldsymbol{\text{line}}\).
  • Problem 19: A computer screen models a \(\boldsymbol{\text{plane}}\).