# Question Video: Finding the Planes That Pass through Given Points Mathematics • 11th Grade

Find three planes that pass through both of the points π΅ and πΆ.

03:10

### Video Transcript

Find three planes that pass through both of the points π΅ and πΆ.

A plane is a space that extends infinitely in all directions and is a set of all points in three dimensions. Hereβs an example of a plane. We will call this plane plane πΎπΏπ. We label a plane by three points that are found not on the same line. Letβs consider the shape we were given.

We need a plane that passes through point π΅ and πΆ. Since we know that a plane needs to be named by three points, we can choose a third point like point π·. And we would say that there is a plane that includes point, π΅, πΆ, and π·. This plane would include the face that is the bottom of this rectangular prism. It would include the face π΄π΅πΆπ·.

But our goal is to find three different planes. So we need to consider another third point that would form a plane with the points π΅ and πΆ. We could try π΅, πΆ, and πΆ prime. They lie on the same plane. We could label this plane as plane π΅πΆπΆ prime. Itβs the plane that includes the front face of this rectangular prism, the face π΅πΆπΆ prime π΅ prime.

What about a third plane? Again, weβll need the points π΅ and πΆ. This one might not seem as immediately obvious. But π΅, πΆ, and π· prime also fall in a plane. You can kind of imagine that this plane runs as a diagonal through our rectangular prism. We will call this one π΅πΆπ· prime. And so we have a list of three planes that pass through the points π΅ and πΆ.

Now letβs go back to the plane π΅πΆπ·. Thatβs the yellow plane that includes the base of the rectangular prism. Remember that we just need three points to name this plane. And so we could also call it π΅πΆπ΄ or π·π΄π΅. If we named it π·π΄π΅, it would still be a correct answer, as the plane π·π΄π΅ includes the points π΅ and πΆ.

In fact, this is true for all of our planes. We could name them in many different ways. We could name the plane that includes the front face π΅π΅ prime πΆ prime. We could name the diagonal plane that cuts through our prism π΄ prime π· prime πΆ. So long as we include three points from that plane.