SOLUTION: Unit 1 Points Lines and Planes Geometry Basics Worksheet ... - Free Printable

Problem Analysis and Solution

The worksheet focuses on identifying points, lines, planes, and their relationships in geometric diagrams. Let's solve each section step by step.

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#### Section 1: Diagram Analysis (First Figure)

The first diagram is a simple geometric figure with labeled points and lines.

Questions:

1. a) How many points appear in the figure?
- The points are labeled as \( D, X, Y, W, V \).
- Answer: 5

2. b) How many lines appear in the figure?
- The lines are:
- Line \( a \) (passing through \( V \) and extending)
- Line \( VW \)
- Line \( WX \)
- Line \( XY \)
- Line \( YD \)
- Line \( DX \)
- Total: 6 lines.
- Answer: 6

3. c) How many planes appear in the figure?
- The figure shows a single plane containing all the points \( D, X, Y, W, V \).
- Answer: 1

4. d) Name a line containing point \( V \).
- Point \( V \) lies on line \( a \) and line \( VW \).
- Answer: Line \( a \)

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#### Section 2: Diagram Analysis (Second Figure)

The second diagram is a more complex geometric figure, likely a rectangular prism or a similar 3D shape, with multiple intersecting lines and planes.

Questions:

1. a) How many points appear in the figure?
- The points are labeled as \( A, B, C, D, E, F, G, H \).
- Answer: 8

2. b) How many lines appear in the figure?
- Counting all the edges of the rectangular prism:
- Front face: \( AB, BC, CD, DA \)
- Back face: \( EF, FG, GH, HE \)
- Connecting edges: \( AE, BF, CG, DH \)
- Total: 12 lines.
- Answer: 12

3. c) How many planes appear in the figure?
- The rectangular prism has 6 faces, each of which is a plane:
- Front face: \( ABCD \)
- Back face: \( EFGH \)
- Top face: \( AEFD \)
- Bottom face: \( BHGC \)
- Left face: \( ADHE \)
- Right face: \( BCGF \)
- Answer: 6

4. d) Name four collinear points.
- Collinear points lie on the same line. For example:
- Points \( A, B, C, D \) are collinear along the front edge of the prism.
- Answer: \( A, B, C, D \)

5. e) Name two non-collinear points.
- Non-collinear points do not lie on the same line. For example:
- Points \( A \) and \( E \) are not collinear.
- Answer: \( A, E \)

6. f) Give another name to line \( c \).
- Line \( c \) appears to be the line passing through points \( B \) and \( F \).
- Another name could be \( BF \).
- Answer: \( BF \)

7. g) Name the intersection of planes \( P \) and \( Q \).
- Planes \( P \) and \( Q \) intersect along a line. From the diagram, this line appears to be \( BD \).
- Answer: Line \( BD \)

8. h) Name the intersection of plane \( R \) and line \( c \).
- Plane \( R \) intersects line \( c \) at a single point. From the diagram, this point is \( B \).
- Answer: Point \( B \)

9. i) Give another name to plane \( L \).
- Plane \( L \) can be named using any three non-collinear points lying on it. For example, if \( L \) is the front face, it can be named as plane \( ABCD \).
- Answer: Plane \( ABCD \)

10. j) Give another name to plane \( P \).
- Plane \( P \) can be named using any three non-collinear points lying on it. For example, if \( P \) is the top face, it can be named as plane \( AEFD \).
- Answer: Plane \( AEFD \)

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#### Section 3: Diagram Analysis (Third Figure)

The third diagram is a triangular prism with labeled vertices.

Questions:

1. a) How many points appear in the figure?
- The points are labeled as \( A, B, C, D, E, F \).
- Answer: 6

2. b) How many lines appear in the figure?
- Counting all the edges of the triangular prism:
- Base triangle: \( AB, BC, CA \)
- Top triangle: \( DE, EF, FD \)
- Connecting edges: \( AD, BE, CF \)
- Total: 9 lines.
- Answer: 9

3. c) How many planes appear in the figure?
- The triangular prism has 5 faces, each of which is a plane:
- Two triangular faces: \( ABC \) and \( DEF \)
- Three rectangular faces: \( ADE, BCF, CEF \)
- Answer: 5

4. d) Name four collinear points.
- Collinear points lie on the same line. For example:
- Points \( A, D, E \) are collinear along one edge.
- Answer: \( A, D, E \)

5. e) Name four coplanar points.
- Coplanar points lie on the same plane. For example:
- Points \( A, B, C, D \) are coplanar on the base plane.
- Answer: \( A, B, C, D \)

6. f) Name the intersection of planes \( ABC \) and \( ADE \).
- The intersection of these two planes is the line \( AD \).
- Answer: Line \( AD \)

7. g) Name the intersection of planes \( BCE \) and \( CEF \).
- The intersection of these two planes is the line \( CE \).
- Answer: Line \( CE \)

8. h) Name the intersection of \( AD \) and \( CF \).
- Lines \( AD \) and \( CF \) intersect at a single point, which is \( C \).
- Answer: Point \( C \)

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Final Answers:

1. Section 1:
- a) 5
- b) 6
- c) 1
- d) Line \( a \)

2. Section 2:
- a) 8
- b) 12
- c) 6
- d) \( A, B, C, D \)
- e) \( A, E \)
- f) \( BF \)
- g) Line \( BD \)
- h) Point \( B \)
- i) Plane \( ABCD \)
- j) Plane \( AEFD \)

3. Section 3:
- a) 6
- b) 9
- c) 5
- d) \( A, D, E \)
- e) \( A, B, C, D \)
- f) Line \( AD \)
- g) Line \( CE \)
- h) Point \( C \)

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Final Answer Boxed:
\[
\boxed{
\begin{array}{l}
\text{Section 1:} \\
\text{a) 5} \\
\text{b) 6} \\
\text{c) 1} \\
\text{d) Line } a \\
\\
\text{Section 2:} \\
\text{a) 8} \\
\text{b) 12} \\
\text{c) 6} \\
\text{d) } A, B, C, D \\
\text{e) } A, E \\
\text{f) } BF \\
\text{g) Line } BD \\
\text{h) Point } B \\
\text{i) Plane } ABCD \\
\text{j) Plane } AEFD \\
\\
\text{Section 3:} \\
\text{a) 6} \\
\text{b) 9} \\
\text{c) 5} \\
\text{d) } A, D, E \\
\text{e) } A, B, C, D \\
\text{f) Line } AD \\
\text{g) Line } CE \\
\text{h) Point } C \\
\end{array}
}
\]