Question

1. use the diagram to the right to name the following.

a) four collinear points.
b) a line that contains point m.
c) a line that contains points h and k.
d) another name for line q.
e) the intersection of lines p and r.

1. use the diagram to the right to name the following.

a) a line containing point f.
b) another name for line k.
c) a plane containing point a.
d) an example of three non - collinear points.
e) the intersection of plane m and line k.

1. use the diagram to the right to name the following.

a) three coplanar points.
b) a plane containing point x.
c) the intersection of plane r and plane zvy.
d) how many planes appear in the figure?
e) how many planes contain point w?

1. use the diagram to the right to name the following.

a) the intersection of lines l and m.
b) another name for plane q.
c) are points d and e collinear or coplanar?
d) how many times do planes p and q intersect?

Explanation:

Step1: Identify collinear points

Points on the same line are collinear. In the first - diagram, points $M$, $N$, $O$, $P$ are on line $p$, so four collinear points are $M$, $N$, $O$, $P$.

Step2: Find line containing point $M$

From the first - diagram, line $p$ contains point $M$.

Step3: Find line containing points $H$ and $K$

In the first - diagram, line $r$ contains points $H$ and $K$.

Step4: Rename line $q$

Line $q$ can also be named as line $OL$ (since two points determine a line).

Step5: Find intersection of lines $p$ and $r$

In the first - diagram, the intersection of lines $p$ and $r$ is point $K$.

Step6: For the second - diagram, find line containing point $F$

Line $AF$ (or $FA$) contains point $F$.

Step7: Rename line $k$

Line $k$ can also be named as line $BC$ (or $CB$).

Step8: Find plane containing point $A$

Plane $M$ contains point $A$.

Step9: Find non - collinear points

In the second - diagram, points $A$, $D$, $F$ are non - collinear.

Step10: Find intersection of plane $M$ and line $k$

The intersection of plane $M$ and line $k$ is point $E$.

Step11: For the third - diagram, find coplanar points

Points $V$, $W$, $X$ are coplanar (lying on plane $R$).

Step12: Find plane containing point $X$

Plane $R$ contains point $X$.

Step13: Find intersection of planes

The intersection of plane $R$ and plane $ZVY$ is line $ZY$.

Step14: Count planes

In the third - diagram, 4 planes appear: plane $R$, plane $ZVY$, plane $ZVX$, plane $ZWX$.

Step15: Count planes containing point $W$

In the third - diagram, 3 planes contain point $W$: plane $R$, plane $ZWX$, plane $ZVY$.

Step16: For the fourth - diagram, find intersection of lines

The intersection of lines $l$ and $m$ is point $E$.

Step17: Rename plane $Q$

Plane $Q$ can also be named as plane $EFG$ (since three non - collinear points determine a plane).

Step18: Check collinearity and coplanarity

Points $D$ and $E$ are non - collinear (not on the same line) but coplanar (both can be in a plane that can be formed to contain both of them).

Step19: Find intersection of planes

Planes $P$ and $Q$ are parallel, so they intersect 0 times.

Answer:

1.
a) $M$, $N$, $O$, $P$
b) line $p$
c) line $r$
d) line $OL$
e) point $K$
2.
a) line $AF$
b) line $BC$
c) plane $M$
d) $A$, $D$, $F$
e) point $E$
3.
a) $V$, $W$, $X$
b) plane $R$
c) line $ZY$
d) 4
e) 3
4.
a) point $E$
b) plane $EFG$
c) coplanar
d) 0