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Question Video: Modeling Points, Lines, and Planes Mathematics • 11th Grade

There are two parallel planes, 𝐴𝐡𝐷 and 𝐸𝐹𝐺. A line 𝑙 intersects the planes 𝐴𝐡𝐷 and 𝐸𝐹𝐺 at points 𝐴 and 𝐺 respectively. The plane 𝐴𝐡𝐷 contains the line segment 𝐴𝐢. Which figure matches the description?

04:44

Video Transcript

There are two parallel planes, 𝐴𝐡𝐷 and 𝐸𝐹𝐺. A line 𝑙 intersects the planes 𝐴𝐡𝐷 and 𝐸𝐹𝐺 at points 𝐴 and 𝐺, respectively. The plane 𝐴𝐡𝐷 contains the line segment 𝐴𝐢. Which figure matches the description?

Although we are given three different options here for this description, let’s see if we can model this scenario ourselves. The first thing we are told is that there are two parallel planes. Parallel planes are usually illustrated as though we were starting to draw a cuboid or rectangular prism, except we don’t include the vertical lines between that prism. This is what we can see in each of the three answer options.

Next, we’re told that the planes are called 𝐴𝐡𝐷 and 𝐸𝐹𝐺. There are some different ways in which we could name a plane. For example, we could call it a single letter, for example, a plane called plane P. Alternatively, since we know that three noncollinear points define a plane, then if we know that a plane contains the points 𝑋, π‘Œ, and 𝑍, then we could call it the plane π‘‹π‘Œπ‘. So, as the first plane is called 𝐴𝐡𝐷, we can assume that there are points 𝐴, 𝐡, and 𝐷 on this plane. Plane 𝐸𝐹𝐺 would contain the points 𝐸, 𝐹, and 𝐺.

At this point, if we looked at the given answer options, we could eliminate some of the possibilities. However, let’s continue with modeling this description.

The next thing we are told is that there is a line 𝑙 which intersects the two planes 𝐴𝐡𝐷 and 𝐸𝐹𝐺. And it intersects them at the points 𝐴 and 𝐺, respectively. So, if we were modeling this, we would start with our line coming through 𝐴. A dotted part of the line indicates the portion of the line we can’t see. The line continues to the second plane and then through to the other side of the second plane 𝐸𝐹𝐺. This line 𝑙 has intersected both planes at the points 𝐴 and 𝐺.

Finally, we are told that the plane 𝐴𝐡𝐷 contains the line segment 𝐴𝐢. So that means there must be another point 𝐢 on plane 𝐴𝐡𝐷 and a line segment connecting these two points. So we can now look at the answer options that we were given. Even if we just use the fact that one of the planes is plane 𝐴𝐡𝐷, we could see in the first two answer options of (A) and (B) that 𝐴, 𝐡, and 𝐷 are not contained on one plane. So these two answer options could not be correct.

Furthermore, if we look at answer option (A), there is a line which passes through 𝐴 and 𝐺. But 𝐴 and 𝐺 don’t lie on either of the two given planes. Answer option (A) also doesn’t have the line segment of 𝐴𝐢. In answer option (B), the line which passes through 𝐴 and 𝐺 is all contained on this lower plane but not intersecting the two planes.

We can then look at answer option (C), which we think is the correct answer option. There are several differences between this answer option and the one that we drew ourselves. So let’s see if the description still matches. Well, firstly, we have the three points 𝐴, 𝐡, and 𝐷 on one plane and the other three points 𝐸, 𝐹, and 𝐺 on the second plane. We then have a line 𝑙 which does pass through 𝐴 and 𝐺 and intersects these. The difference is that we can see that there is a right angle drawn. This just means that, in fact, we know that the line 𝑙 must intersect these planes orthogonally at right angles. This is an extra piece of information on this diagram, but it would still be correct that the line 𝑙 intersects these two planes.

And finally we can also see that there is indeed a line segment 𝐴𝐢 on this plane 𝐴𝐡𝐷. It doesn’t matter that in this figure the line segment 𝐴𝐢 also lies on the line 𝐴𝐡. It still matches the description that we were given. Therefore, we can give the answer that it is the figure in option (C) which matches the given description.

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