Consider the following figure. Write another name for the plane 𝑝. How many planes are shown in the figure? Which two planes intersect in the line segment 𝐴𝐷? Are planes 𝑝 and 𝐷𝐸𝐹 parallel or intersecting?
Let’s begin by considering this plane 𝑝. We can recall that a plane is a two-dimensional surface made up of points that extends infinitely in all directions and there exists exactly one plane through any three noncollinear points. This word, noncollinear, simply means that the three points don’t lie on a straight line. And it is this second sentence in this statement that will help us answer the first part of this question to write another name for the plane 𝑝.
Well, because we know that there is one plane through any three noncollinear points, then we can also define a plane as the plane which passes through three given points. In this figure, we can see that the points 𝐴, 𝐵, and 𝐶 lie on plane 𝑝. Therefore, another name that we could give to plane 𝑝 could be plane 𝐴𝐵𝐶. Of course, plane 𝐵𝐴𝐶 or plane 𝐶𝐵𝐴 would also be valid answers.
Now, let’s have a look at the second part of this question to identify how many planes there are in this figure. We might think that this is confusing as surely there’s just one plane, the plane 𝑝, or plane 𝐴𝐵𝐶. However, using the fact that a plane is a two-dimensional surface means that we can identify more planes in this figure. We can see, shaded in orange, that there would be another plane at the back of this figure. We could use the points which lie on this plane to identify it as the plane 𝐴𝐶𝐹𝐷. We can then also identify the plane which passes through the points 𝐴, 𝐵, 𝐸, and 𝐷. This would be plane 𝐴𝐵𝐸𝐷. Then we have another plane passing through the points 𝐵, 𝐶, 𝐹, 𝐸 and finally one more plane which passes through the three points at the top of this figure, giving us plane 𝐷𝐸𝐹.
Combined with plane 𝑝 or plane 𝐴𝐵𝐶, that would give us five different planes. As there are no other planes on this figure, we can give the answer for the second part of the question that there are five planes shown in the figure.
We can then look at the third part of this question which asks, which two planes intersect in the line segment 𝐴𝐷? Let’s identify that the line segment 𝐴𝐷 is marked here on the figure in pink. And we will have two planes which intersect here at the line segment 𝐴𝐷. We’ll have this plane 𝐴𝐶𝐹𝐷 and this plane 𝐴𝐵𝐸𝐷. There are no other planes shown in this figure which intersect in the line segment 𝐴𝐷. So the answer to this third part is plane 𝐴𝐶𝐹𝐷 and plane 𝐴𝐵𝐸𝐷.
And finally, let’s take a look at the fourth part of this question which asks, are planes 𝑝 and 𝐷𝐸𝐹 parallel or intersecting? The plane 𝐷𝐸𝐹 will lie at the top of this figure. Let’s recall that there are three possible relationships that can exist between two planes in space. Firstly, they can be parallel. Two parallel planes do not intersect. Secondly, two planes can be intersecting. As we saw in the previous part of this question, when two planes intersect, the intersection is a line. And thirdly, although we are not asked about it in this part of the question, two planes can be coincident. That means that the two planes would share all the same points.
So let’s look at the figure and see if we can identify if planes 𝑝 and 𝐷𝐸𝐹 are parallel or intersecting. Now we might look at this diagram and visualize it as a three-dimensional object. We might visualize this as a triangular prism, in which case it’s very likely that the top and the lower base would be parallel. However, we have to be really careful here because actually we don’t have enough information to say that for sure. We don’t know, for example, that the length 𝐶𝐹 is the same as the length 𝐴𝐷 is the same as the length 𝐵𝐸. If these three lengths or distances were all the same, then the two planes would be parallel. If the planes are not parallel, then they would be intersecting. But we don’t know for sure. So the statement for the answer needs to reflect this.
We can therefore give the answer for the fourth and final part of this question as “It cannot be determined without more information.”