Intersecting Lines Coplanar?Date: 08/23/2002 at 10:19:53 From: Alice Matthews Subject: Geometry - Intersecting lines Do two intersecting lines have to be coplanar? The definition in our Geometry book for intersecting lines states that two intersecting lines must be coplanar, but we can't understand why they couldn't lie in more than one plane and still intersect one another. Mrs. Matthews' 8th grade Geometry class Date: 08/23/2002 at 16:18:19 From: Doctor Rick Subject: Re: Geometry - Intersecting lines Hi, Alice. Each line is in more than one plane. For instance, a vertical line is in lots of different vertical planes. Thus you could say that one line is in plane A and the other line is in a different plane B, and the two can still intersect. For instance, you could have two horizontal lines that intersect, and say that one line is in one vertical plane and the other line is in a different vertical plane. That's not the point. The point is that if the lines intersect, there must be ONE plane that BOTH lines are in. The two horizontal lines are in the SAME horizontal plane, even though they are in different vertical planes. Has the class learned that three points determine a unique plane? Pick one point at the intersection of the lines, and pick one other point on each line. These three points determine a plane. Since two points on each line are in that plane, both entire lines must lie in that plane. Two lines that are not coplanar can be pictured as the center line of a road and the center line of another road that passes above the first on a bridge. They never intersect. Each is in a horizontal plane, but the planes are parallel, one above the other. The closest that the lines ever get to one another is the distance between the planes (the height of the bridge). In order to get the lines to intersect, you have to reduce the height of the bridge to zero - and then the planes aren't parallel any more; they are the same plane! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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