### Video Transcript

Consider the following figure. Write another name for the plane π. How many planes are shown in the figure? Which two planes intersect in the line segment π΄π·? Are planes π and π·πΈπΉ parallel or intersecting?

Letβs begin by considering this plane π. 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 donβt 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 π.

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 π΄, π΅, and πΆ lie on plane π. Therefore, another name that we could give to plane π could be plane π΄π΅πΆ. Of course, plane π΅π΄πΆ or plane πΆπ΅π΄ would also be valid answers.

Now, letβs 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 thereβs just one plane, the plane π, or plane π΄π΅πΆ. 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 π΄πΆπΉπ·. We can then also identify the plane which passes through the points π΄, π΅, πΈ, and π·. This would be plane π΄π΅πΈπ·. Then we have another plane passing through the points π΅, πΆ, πΉ, πΈ and finally one more plane which passes through the three points at the top of this figure, giving us plane π·πΈπΉ.

Combined with plane π or plane π΄π΅πΆ, 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 π΄π·? Letβs identify that the line segment π΄π· is marked here on the figure in pink. And we will have two planes which intersect here at the line segment π΄π·. Weβll have this plane π΄πΆπΉπ· and this plane π΄π΅πΈπ·. There are no other planes shown in this figure which intersect in the line segment π΄π·. So the answer to this third part is plane π΄πΆπΉπ· and plane π΄π΅πΈπ·.

And finally, letβs take a look at the fourth part of this question which asks, are planes π and π·πΈπΉ parallel or intersecting? The plane π·πΈπΉ will lie at the top of this figure. Letβs 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 letβs look at the figure and see if we can identify if planes π and π·πΈπΉ 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 itβs very likely that the top and the lower base would be parallel. However, we have to be really careful here because actually we donβt have enough information to say that for sure. We donβt know, for example, that the length πΆπΉ is the same as the length π΄π· is the same as the length π΅πΈ. 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 donβt 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 βIt cannot be determined without more information.β