Question 48997: Can you please help me with this problem? My math teacher is out because his daughter has cancer so he took the week out and this is only the SECOND week of school. So can you please help me with this one question then i may be able to complete it on my on.
Determine if the conjecture is true or false based on the given information. Explain your answer and give a counter example for any false conjecture.
Given: noncollinear points A, B, C, and D
Conjecture: A, B, C, and D are coplanar.
Answer by Born2TeachMath(20)   (Show Source):
You can put this solution on YOUR website! The conjecture is False.
We're given that A, B, C, and D are non-colinear. That just means that they are not ALL on the SAME LINE. (More here later!) What we're thinking about is do they all lie on the SAME PLANE, like a piece of paper?
You probably know that it takes 2 points to define a line, especially if you've had algebra before. Think about it: If you draw two points, you can always draw a line through them, no matter what.
It takes a minimum of 3 points to define a plane. Think about a three-legged stool. It always sits flat on the floor, no matter how rough or bumpy the floor might be. Each leg will tough the floor, and the stool will sit still - it won't wobble back and forth.
Now look at a four-legged chair, the bottom of which are points A, B, C, and D. Do all four-legged chairs sit perfectly on the ground, no matter how bumpy? No, of course not! That's because planes are defined by three points, and the fourth point is too many, and might not HAVE to be on the plane of the floor. Therefore the conjecture is false.
The conjecture says that all four points MUST define a single plane, and you can see using the four-legged chair that that does not HAVE to be correct. It MIGHT be, the floor could be perfectly smooth!), but it doesn't have to be!!
One last point. There is another possibility. The Given says that A, B, C, and D are noncollinear. That means that all FOUR do not lie on the same line. However, assume that A, B, and C are collinear, and that D does not. Since A, B, and C are collinear, they really act like only two points (since a line needs only 2 points). Therefore one of the points is unneccessary - say point C. Therefore this problem now becomes only three points A, B, and D. And with three-leggged stools, these three are now coplanar! And since C is extra, it too is coplanar. Therefore A, B, C, and D are now coplanar.
There are no other exceptions to look at - just this one.
However, in the most general sense of the GIVEN, we can not assume that A, B, and C are collinear, while D is not. Therefore, in the most general sense, we can not conclude FOR CERTAIN that A, B, C, and D are all coplanar. They might, be, but they don't HAVE TO BE!
I hope this helps! All of this collinearity and coplanarity can be confusing. Just try to picture four points floating in space, and see if you can slide a pieces of flat paper through them!
Good luck in geometry, and hang in there. It'll get better!
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