Question

date: 8/11/25
per:
homework 1: points, lines, and planes

1. use the diagram to answer the following questions.

a) how many points appear in the figure?
b) how many lines appear in the figure?
c) how many planes appear in the figure?
d) name a line containing point v.
e) name the intersection of lines a and b.
f) give another name for line b.
g) name three non - collinear points.
h) give another name for plane d.

1. use the diagram to answer the following questions.

a) how many points appear in the figure?
b) how many lines appear in the figure?
c) how many planes appear in the figure?
d) name three collinear points.
e) name four non - coplanar points.
f) give another name for line e.
g) name the intersection of (overrightarrow{pq}) and (overrightarrow{mo}).
h) name the intersection of plane k and line c.
i) give another name for plane l.
j) give another name for (overrightarrow{pq}).

1. use the diagram to answer the following questions.

a) how many points appear in the figure?
b) how many lines appear in the figure?
c) how many planes appear in the figure?
d) name three collinear points.
e) name four coplanar points.
f) name the intersection of planes abc and abe.
g) name the intersection of planes bch and def.
h) name the intersection of (overline{ad}) and (overline{df}).

Explanation:

1. First diagram (points \(V, W, X, Y, Z\))

Step1: Count points

By observation, there are 5 points (\(V, W, X, Y, Z\)).

Step2: Count lines

Lines are \(a\), \(b\), \(\overleftrightarrow{XY}\), \(\overleftrightarrow{YW}\), \(\overleftrightarrow{WX}\), so 5 lines.

Step3: Count planes

There is 1 plane labeled \(D\).

Step4: Name line with point \(V\)

Line \(a\) contains point \(V\).

Step5: Inter - section of \(a\) and \(b\)

The intersection of lines \(a\) and \(b\) is point \(W\).

Step6: Another name for line \(b\)

Line \(b\) can be named \(\overleftrightarrow{YZ}\) (or \(\overleftrightarrow{YW}\), \(\overleftrightarrow{WZ}\)).

Step7: Non - collinear points

For example, \(V\), \(X\), \(Y\) are non - collinear.

Step8: Another name for plane \(D\)

Plane \(D\) can be named plane \(VWXYZ\) (using all the points on the plane).

2. Second diagram (points \(M, N, O, P, Q, R, S, T, U\))

Step1: Count points

There are 9 points.

Step2: Count lines

Lines are \(c\), \(d\), \(e\), \(\overleftrightarrow{PQ}\), \(\overleftrightarrow{MO}\), \(\overleftrightarrow{RS}\), so 6 lines.

Step3: Count planes

There are 2 planes (\(K\) and \(L\)).

Step4: Collinear points

\(M\), \(N\), \(O\) are collinear.

Step5: Non - coplanar points

For example, \(P\), \(T\), \(M\), \(U\) are non - coplanar.

Step6: Another name for line \(e\)

Line \(e\) can be named \(\overleftrightarrow{MO}\).

Step7: Inter - section of \(\overleftrightarrow{PQ}\) and \(\overleftrightarrow{MO}\)

The intersection of \(\overleftrightarrow{PQ}\) and \(\overleftrightarrow{MO}\) is point \(N\).

Step8: Inter - section of plane \(K\) and line \(c\)

The intersection of plane \(K\) and line \(c\) is point \(R\).

Step9: Another name for plane \(L\)

Plane \(L\) can be named plane \(MNOPU\) (using some of the points on the plane).

Step10: Another name for \(\overleftrightarrow{PQ}\)

\(\overleftrightarrow{PQ}\) can be named \(\overleftrightarrow{PNQ}\) (using points on the line).

3. Third diagram (points \(A, B, C, D, E, F, G, H\))

Step1: Count points

There are 8 points.

Step2: Count lines

Lines are \(\overleftrightarrow{AB}\), \(\overleftrightarrow{BC}\), \(\overleftrightarrow{CD}\), \(\overleftrightarrow{DA}\), \(\overleftrightarrow{AE}\), \(\overleftrightarrow{BF}\), \(\overleftrightarrow{CG}\), \(\overleftrightarrow{DH}\), \(\overleftrightarrow{EF}\), \(\overleftrightarrow{FG}\), \(\overleftrightarrow{GH}\), \(\overleftrightarrow{HE}\), so 12 lines.

Step3: Count planes

There are 3 planes (e.g., plane \(ABC\), plane \(DEF\), plane \(ABFE\)).

Step4: Collinear points

There are no three collinear points shown.

Step5: Coplanar points

For example, \(A\), \(B\), \(C\), \(D\) are coplanar.

Step6: Inter - section of planes \(ABC\) and \(ABE\)

The intersection of planes \(ABC\) and \(ABE\) is line \(\overleftrightarrow{AB}\).

Step7: Inter - section of planes \(BCH\) and \(DEF\)

There is no intersection shown (assuming they are parallel).

Step8: Inter - section of \(\overleftrightarrow{AD}\) and \(\overleftrightarrow{DF}\)

The intersection of \(\overleftrightarrow{AD}\) and \(\overleftrightarrow{DF}\) is point \(D\).

Answer:

1.
a) 5
b) 5
c) 1
d) Line \(a\)
e) \(W\)
f) \(\overleftrightarrow{YZ}\)
g) \(V\), \(X\), \(Y\)
h) Plane \(VWXYZ\)
2.
a) 9
b) 6
c) 2
d) \(M\), \(N\), \(O\)
e) \(P\), \(T\), \(M\), \(U\)
f) \(\overleftrightarrow{MO}\)
g) \(N\)
h) \(R\)
i) Plane \(MNOPU\)
j) \(\overleftrightarrow{PNQ}\)
3.
a) 8
b) 12
c) 3
d) None
e) \(A\), \(B\), \(C\), \(D\)
f) \(\overleftrightarrow{AB}\)
g) None
h) \(D\)