For P to be collinear with two points, a line through these points must contain P.
B
b
Point J lies on the edge between the ceiling and the wall on the left-hand side.
C
c
In how many different planes are the points in the diagram contained?
A
a
K and N
B
b
Example Solution: JKL and JKP
C
c
J, K, L, M, N, P, and Q
Practice makes perfect
a
We are looking for two points that are collinear with P. If we draw a line through these two points, P must also lie on that line. In the diagram, there is only one such line, KN.
Points K and N are, therefore, the only two points that are collinear with P.
b
Point J lies on the edge between the ceiling and the wall on the left-hand side. Therefore, J is contained in the two planes that are extensions of the ceiling and this wall, respectively. To name these two planes, we have to find three points that are:
not collinear and
are in each plane.The ceiling contains points J, K, and L. Thus, we can name the first plane JKL.
The wall on the left-hand side contains points J, K, P, N, and Q. Therefore, there are multiple ways of naming this second plane using point P — any combination of those points would work. Let's arbitrarily name it JKP.
c
If a point lies on the edge between two planes, it is in two of the planes identifiable in the diagram. If a point instead lies in a corner, where three planes meet, that point is instead in three of the planes of the diagram. All of the points in the diagram are either on an edge or in a corner. Thus, all of them are in more than one plane — points J, K, L, M, N, P, and Q are in more than one plane.
