Chapter 5.4, Problem 37PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Determine whether each statement about the shortest distance between a point and a line or plane can be proved using a direct or indirect proof.

Given: $\overline{AB}\perp$ line $p$

Prove: $\overline{AB}$ is the shortest segment from $A$ to line $p$ .

  

Calculation:

In an indirect proof or proof by contradiction, assume that the conclusion is false. By showing the assumption to be logically impossible, prove the assumption is false and the original conclusion is true. For this problem, assume that $\overline{AB}$ is not the shortest segment from $A$ to $p$ and prove it false.

Given: $\overline{AB}\perp$ $p$

Prove: $\overline{AB}$ is the shortest segment from $A$ to $p$ .

  

Assume $\overline{AB}$ is not the shortest segment from $A$ to $p$ .

Since $\overline{AB}$ is not the shortest segment from $A$ to $p$ , there is a point $C$ such that $\overline{AC}$ is the shortest distance. $\angle ABC$ is a right triangle with hypotenuse $\overline{AC}$ , the longest side of $\angle ABC$ since it is across from the largest angle in $\angle ABC$ by the Angle- Side Relationships in Triangle Theorem.

This contradicts the fact that $\overline{AC}$ is the shortest side. Therefore, the assumption is false, and the conclusion, $\overline{AB}$ is the shortest side, must be true.