Chapter 10.5, Problem 5ST2

To determine

To construct: a segment $\overline{PQ}$ , whose length is the geometric mean of $AB$ and $AC$ .

Explanation of Solution

Construct a segment of length $x$ with the condition as follows;

  $x=\sqrt{AC\times AB}$

Draw a line $RS=\overline{AB}$ and $ST=\overline{AC}$ .

Locate $O$ the midpoint of $RT$ by constructing perpendicular bisector.

Draw a semicircle taking $O$ as center and $OR$ as radius.

Draw a perpendicular to $RT$ on $S$ and name the point where it intersects the semicircle as $Z$ .

Therefore, $ZS=x$ is equal to $\sqrt{AC\times AB}$ .