Question Video: Finding the Intersection of Two Planes Mathematics • 11th Grade

Video Transcript

𝑋 and 𝑌 are planes that intersect at line 𝐿. 𝐵 is a point on plane 𝑋, and 𝐶 is a point on plane 𝑌. Determine the intersection of plane 𝑌 with plane 𝐴𝐵𝐶.

In our figure, we can see line 𝐿 is the intersections of plane 𝑋 and 𝑌. We want the intersection of plane 𝑌 with plane 𝐴𝐵𝐶. Let’s first identify plane 𝑌. We can see that the plane 𝑌 includes the points 𝐴 and 𝐶. And if we’re interested in plane 𝐴𝐵𝐶, we recognize that the points 𝐴 and 𝐵 are located along the same plane, plane 𝑋, and the points 𝐶 and 𝐴 are both located in the plane 𝑌. This tells us that the point 𝐴 is located in plane 𝑋 and in plane 𝑌.

Because the points 𝐴, 𝐵, and 𝐶 are noncolinear, they can form exactly one plane. To see plane 𝐴𝐵𝐶, first, we’ll connect a line through points 𝐴 and 𝐵 and another line through the points 𝐴, 𝐶. From there, we can add an additional line that goes through point 𝐶. By connecting these lines, we have some idea of how this plane would look.

At this point, we should start to see that the plane 𝐴𝐵𝐶 is slicing through the plane 𝑌. And this slicing is happening along the line that goes through the points 𝐴 and 𝐶. Plane 𝑌 contains the line 𝐴𝐶, as does plane 𝐴𝐵𝐶, which makes the intersection of plane 𝑌 and plane 𝐴𝐵𝐶 the line 𝐴𝐶.