Ch 1 Worksheet Points, Lines and Planes 1. Refer to the diagram: a ... - Free Printable

Problem 1: Refer to the diagram

#### a) Name 2 planes that intersect in HG.
- The line segment \( HG \) is part of the bottom face of the rectangular prism.
- The two planes that intersect along \( HG \) are:
- The bottom face of the prism (plane \( EFGH \)).
- The right face of the prism (plane \( BCGH \)).

Answer: Planes \( EFGH \) and \( BCGH \).

#### b) Are the points A, B, C, and D collinear?
- Collinear points lie on the same straight line.
- Points \( A \), \( B \), \( C \), and \( D \) do not lie on the same straight line. They form the vertices of the top face of the rectangular prism, which is a quadrilateral.

Answer: No.

#### c) Are the points A, B, C, and D coplanar?
- Coplanar points lie on the same plane.
- Points \( A \), \( B \), \( C \), and \( D \) all lie on the top face of the rectangular prism, which is a single plane.

Answer: Yes.

#### d) Name 2 planes that do not intersect.
- In a rectangular prism, opposite faces are parallel and do not intersect.
- Two such planes are:
- The top face (plane \( ABCD \)).
- The bottom face (plane \( EFGH \)).

Answer: Planes \( ABCD \) and \( EFGH \).

#### e) Name 3 lines that intersect at C.
- Point \( C \) is a vertex of the rectangular prism. Three lines that intersect at \( C \) are:
- Line \( BC \) (connecting \( B \) and \( C \)).
- Line \( CG \) (connecting \( C \) and \( G \)).
- Line \( CD \) (connecting \( C \) and \( D \)).

Answer: Lines \( BC \), \( CG \), and \( CD \).

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Problem 2:

#### a) The ray opposite to \( \overrightarrow{KN} \) is _________.
- The ray \( \overrightarrow{KN} \) starts at point \( K \) and extends through \( N \).
- The ray opposite to \( \overrightarrow{KN} \) starts at point \( N \) and extends through \( K \). This is \( \overrightarrow{NK} \).

Answer: \( \overrightarrow{NK} \).

#### b) Another name for \( \overline{LM} \) is _________.
- The segment \( \overline{LM} \) can also be named by reversing the order of the endpoints. Thus, another name is \( \overline{ML} \).

Answer: \( \overline{ML} \).

#### c) \( LN = \) _________ (what value?).
- From the diagram, point \( L \) is at coordinate 3, and point \( N \) is at coordinate 7.
- The length of \( \overline{LN} \) is the distance between these coordinates:
\[
LN = |7 - 3| = 4
\]

Answer: 4.

#### d) The coordinate of the midpoint of \( \overline{JM} \) is _________.
- Point \( J \) is at coordinate -4, and point \( M \) is at coordinate 5.
- The midpoint formula for a segment with endpoints \( (x_1) \) and \( (x_2) \) is:
\[
\text{Midpoint} = \frac{x_1 + x_2}{2}
\]
- Substituting the coordinates:
\[
\text{Midpoint} = \frac{-4 + 5}{2} = \frac{1}{2} = 0.5
\]

Answer: 0.5.

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Problem 3:

#### a) If \( TE = 5x \) and \( EP = x \), then \( x = \) _________.
- From the diagram, \( T \), \( E \), and \( P \) are collinear points.
- The total length \( TP \) is the sum of \( TE \) and \( EP \):
\[
TP = TE + EP
\]
- Given \( TE = 5x \) and \( EP = x \):
\[
TP = 5x + x = 6x
\]
- From the diagram, the coordinate of \( T \) is -3, and the coordinate of \( P \) is 4. Thus:
\[
TP = |4 - (-3)| = 4 + 3 = 7
\]
- Equating the two expressions for \( TP \):
\[
6x = 7 \implies x = \frac{7}{6}
\]

Answer: \( \frac{7}{6} \).

#### b) The coordinate of \( E \) is _________.
- The coordinate of \( E \) can be found using the fact that \( TE = 5x \) and \( x = \frac{7}{6} \):
\[
TE = 5x = 5 \cdot \frac{7}{6} = \frac{35}{6}
\]
- Since \( T \) is at -3, the coordinate of \( E \) is:
\[
\text{Coordinate of } E = -3 + \frac{35}{6} = \frac{-18}{6} + \frac{35}{6} = \frac{17}{6}
\]

Answer: \( \frac{17}{6} \).

#### c) If \( T \) is the midpoint of \( \overline{SP} \), find the coordinate of \( S \).
- If \( T \) is the midpoint of \( \overline{SP} \), then the midpoint formula gives:
\[
T = \frac{S + P}{2}
\]
- Given \( T = -3 \) and \( P = 4 \):
\[
-3 = \frac{S + 4}{2}
\]
- Solving for \( S \):
\[
-3 \cdot 2 = S + 4 \implies -6 = S + 4 \implies S = -6 - 4 = -10
\]

Answer: -10.

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Final Answers:

1. a) Planes \( EFGH \) and \( BCGH \).
b) No.
c) Yes.
d) Planes \( ABCD \) and \( EFGH \).
e) Lines \( BC \), \( CG \), and \( CD \).

2. a) \( \overrightarrow{NK} \).
b) \( \overline{ML} \).
c) 4.
d) 0.5.

3. a) \( \frac{7}{6} \).
b) \( \frac{17}{6} \).
c) -10.

\[
\boxed{\text{See detailed solutions above.}}
\]