a How many planes on the diagram contain both the line and point?
B
b How many planes on the diagram contain both the line and point?
C
c Take the two points on the line and the given point, and use Postulate 1-4.
D
d Take the two points on the line and the given point, and use Postulate 1-4.
E
e Think of whether a line and a point not on the line are coplanar.
A
a 1
B
b 1
C
c 1
D
d 1
E
e They are coplanar on one unique plane.
Practice makes perfect
a From the diagram, we can see that the plane EFG contains both the line EF and point G.
We can also see that the plane PEF contains the line, but it does not contain the point. Similarly, the point also lies on the planes PQG and FGQ, but those planes do not contain the line. Therefore, there is only one plane, which contains both the line and point.
b Similar to Part A, we can see that the only plane that contains both the line PH and point E is PHE. There are planes that contain the line or the point, but no more that contain both.
c Looking at the diagram, we cannot see a plane that contains the line FG and the point P. Although, let's look at the situation another way. We have the three points: F, G and P. They do not lie on the same line, so they are noncollinear. Now, let's consider Postulate 1-4.
Through any three noncollinear points
there is exactly one plane.
Therefore, there is exactly one plane through the points F, G and P. Consequently, there is exactly one plane through the line FG and point P. It would need to be drawn as shown below.
This is plane PFG.
d Similar to Part C, there is no plane on the diagram that contains the line EP and the point G. Although, using the same thought process as in Part C, and Postulate 1-4, we can conclude that there is exactly one plane through the line EP and the point G.
e Let's take three noncollinear points. Then, recall what Postulate 1-4 tells us.
Through any three noncollinear points,
there is exactly one plane.
Therefore, there is only one plane through these points. Now, let's think about Postulate 1-1.
Through any two points
there is only one line.
Thus, we can draw a line through the two of our points. This way we get a line and a point that does not lie on that line. Earlier, we concluded that there is only one plane through the points. This means that the line and point are coplanar on one unique plane that passes through the line and the point.
