Question

use the diagram to answer the following questions.

1. plane p and q are parallel.

a) the intersection of lines l and m point e
b) another name for plane q plane gie
c) are points d and e collinear or coplanar? collinear
d) how many times do planes p and q intersect? 0
2.
a) how many points appear in the figure? 9
b) how many lines appear in the figure? 3
c) how many planes appear in the figure? 2
d) name three collinear points. m,n,o
e) name four non - coplanar points p,q,r,t
f) give another name for line e mo
g) name the intersection of pq and mo point n
h) name the intersection of plane k and line c point s
i) give another name for plane t plane mno
j) give another name for pq qp

Explanation:

Step1: Analyze line - line intersection

Lines \(l\) and \(m\) intersect at a single point. From the diagram and given answer, it is Point \(E\).

Step2: Name a plane

A plane can be named by three non - collinear points on it. For plane \(Q\), Points \(G\), \(I\), \(E\) are on it, so it can be named Plane \(GIE\).

Step3: Determine collinearity or coplanarity

Points \(D\) and \(E\) lie on the same line, so they are collinear.

Step4: Analyze plane - plane intersection

Parallel planes do not intersect. Since planes \(P\) and \(Q\) are parallel, they intersect \(0\) times.

Step5: Count points

By observing the figure, we can count that there are 9 points.

Step6: Count lines

By observing the figure, we can count that there are 3 lines.

Step7: Count planes

By observing the figure, we can count that there are 2 planes.

Step8: Find collinear points

Points \(M\), \(N\), \(O\) lie on the same line, so they are collinear.

Step9: Find non - coplanar points

Points \(P\), \(Q\), \(R\), \(T\) do not lie on the same plane, so they are non - coplanar.

Step10: Rename a line

Line \(e\) can also be named \(\overleftrightarrow{MO}\) as it passes through points \(M\) and \(O\).

Step11: Find line - line intersection

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

Step12: Find line - plane intersection

The intersection of plane \(K\) and line \(c\) is Point \(S\).

Step13: Rename a plane

Plane \(t\) can also be named Plane \(MNO\) as it contains points \(M\), \(N\), \(O\).

Step14: Rename a line segment

\(\overline{PQ}\) can also be named \(\overline{QP}\) as a line segment has two endpoints and can be named in reverse order.

Answer:

1.
a. Point \(E\)
b. Plane \(GIE\)
c. Collinear
d. \(0\)
2.
a. \(9\)
b. \(3\)
c. \(2\)
d. \(M\), \(N\), \(O\)
e. \(P\), \(Q\), \(R\), \(T\)
f. \(\overleftrightarrow{MO}\)
g. Point \(N\)
h. Point \(S\)
i. Plane \(MNO\)
j. \(\overline{QP}\)