Chapter 5.4, Problem 38PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $\overline{PQ}\perp$ plane $M$

Prove: $\overline{PQ}$ is the shortest segment from $P$ to plane $M$ .

  

Calculation:

A direct proof is more appropriate for this proof, as the conclusion is directly related to the given information.

Given: $\overline{PQ}\perp$ plane $M$

Prove: $\overline{PQ}$ is the shortest segment from $P$ to plane $M$ .

  

From the definition, $\overline{PQ}$ is perpendicular to plane $M$ if it is perpendicular to every line in $M$ that intersect it. But since the perpendicular segment from a point to a line is the shortest from the point to the line, that perpendicular segment is the shortest segment from the point to each of these lines. Therefore, $\overline{PQ}$ is the shortest segment from $P$ to $M$ .