Determine whether the following statements are always,sometime, or never true.Explain
1.Three points determine a plane.
2.The intersection of two planes can be a point.
1.Never true, the three points must be noncollinear.
2.Never true, the intersection of two planes is a line.
(Are these answers right?) Thanks
1. 3 points do determine a plane, unless the 3 points are collinear, so "sometime"
2. they form a line, unless they are parallel, but never in a point
1. Three points determine a plane. - Sometimes true. If the three points are noncollinear (not lying on the same line), then they will always determine a unique plane. However, if the three points are collinear, meaning they do lie on the same line, then they will not determine a unique plane.
2. The intersection of two planes can be a point. - Sometimes true. If the two planes are distinct and not parallel, then their intersection will be a line. However, if the two planes are coincident or parallel, then their intersection will be either a plane (if coincident) or no intersection at all (if parallel).
Your answers are correct.
1. The statement "Three points determine a plane" is never true. For three points to determine a unique plane, they must be noncollinear, meaning they cannot all lie on the same line. If the three points are collinear, they do not determine a plane.
2. The statement "The intersection of two planes can be a point" is also never true. When two planes intersect, their intersection will always form a line. This is because two distinct planes can either be parallel (resulting in no intersection) or they will intersect along a line. Therefore, the intersection of two planes cannot be a single point.
Yes, your answers are correct.
1. The statement "Three points determine a plane" is never true. In order for three points to determine a plane, they must be noncollinear, which means they cannot all lie on the same line. If the three points are collinear, they do not determine a unique plane because they lie on an infinite number of planes.
2. The statement "The intersection of two planes can be a point" is also never true. When two planes intersect, they do so along a line called the intersection line. It is not possible for the intersection of two planes to be a single point.