Question

Suppose the plane, T, and three collinear (but not coplanar) points, P, Q, and W. Part A: Can either P, Q, or W be a point on T? Justify your answer. Part B: If the lengths of PW and PQ are given, can the length of QW be determined? Justify your answer.

a) Part A: Yes, because collinearity implies coplanarity; Part B: Yes, by the Triangle Inequality Theorem.
b) Part A: No, because collinearity does not imply coplanarity; Part B: Yes, by the Converse of the Triangle Inequality Theorem.
c) Part A: Yes, because collinearity implies coplanarity; Part B: No, because information about the plane is needed.
d) Part A: No, because collinearity does not imply coplanarity; Part B: No, because information about the plane is needed.

Answer 1

The correct answer is option b: Part A is 'No' since collinear points do not have to be in the same plane, and Part B is 'Yes' as the length of QW can be found from PW and PQ because points P, Q, and W lie on the same line.

Step-by-step explanation:

The response to the student's question is as follows:

Part A:

No, the fact that points P, Q, and W are collinear does not necessarily mean that they are coplanar with plane T. Collinearity implies that the points lie on a single straight line, but that line does not have to be within plane T. Therefore, collinearity alone does not ensure that any of the points are on the plane.

Part B:

Yes, if the lengths of PW and PQ are given, the length of QW can be determined. This is because P, Q, and W are specified to be collinear. If we know two side lengths, PW and PQ, on a straight line, we can find the third length, QW, by either adding or subtracting these lengths depending on whether point Q lies between P and W or not.

To summarize, the correct answer is (b): Part A: No, because collinearity does not imply coplanarity; Part B: Yes, by the Converse of the Triangle Inequality Theorem.