Multiple Response: Please select all correct answers and click “submit.” Four points are always coplanar if: Check all that apply. A. they are collinear. B. they lie on different planes. C. they lie on different lines. D. they lie in the same plane.
The Correct Answer and Explanation is:
Correct Answers:
✅ A. they are collinear.
✅ D. they lie in the same plane.
Explanation:
The concept of coplanarity refers to the condition where multiple points lie within the same geometric plane. Four points are always coplanar under specific circumstances.
Option A: They are collinear.
When four points are collinear, they lie along the same straight line. A line itself is a subset of a plane, meaning any line can be contained within a plane. Therefore, if four points are collinear, a single plane can always be found that includes the entire line and hence all four points. This makes collinearity a sufficient condition for coplanarity.
Option D: They lie in the same plane.
This statement directly reflects the definition of coplanarity. If four points lie in the same plane, then by definition, they are coplanar. This is a universally true condition and is not dependent on the arrangement of the points, whether linear or not.
Option B: They lie on different planes.
Points that lie on different planes cannot all belong to a single common plane unless the planes intersect along a line and the points all happen to lie on that line. However, the phrase “different planes” typically implies the absence of a shared plane for all four points. Therefore, this condition does not guarantee coplanarity.
Option C: They lie on different lines.
The fact that points lie on different lines does not provide enough information to determine coplanarity. Different lines can exist in the same plane or in completely separate planes. Without additional information, this condition cannot confirm coplanarity.
Hence, only options A and D ensure that four points are always coplanar.
