Chapter 1.4, Problem 23WE

To determine

To find: the number of lines can draw through $O$ that will form a $30°$ angle with $\overrightarrow{OB}$ .

Answer to Problem 23WE

Only 2 lines can be drawn through point O to form $30°$ with $\overline{OB}$ .

Explanation of Solution

On $\overline{AH}$ in a given plane, choose any point O between A and H. Consider $\overline{OA}$ and $\overline{OH}$ and all the rays than can be drawn from O on one side of $\overline{AH}$ . These rays can be paired with the real numbers from 0 to 180 in such a way that:

The line $\overline{OA}$ is paired with 0, and $\overline{OH}$ with 180.

If $\overline{OB}$ is paired with $x$ , and $\overline{OC}$ with $y$ , then $m\angle BOC=\left|x-y\right|$ .

Thus the figure can be drawn as shown below:

As it can be seen from the above figure it can observe that on the plane of the paper through point O between A and B, there are maximum two lines that will make $30°$ with $\overline{OB}$ .

The line $\overline{OC}$ can be drawn using protractor on one side of the $\overline{AB}$ and $\overline{OD}$ on the other side of the $\overline{AB}$ .

Therefore, only 2 lines can be drawn through point O to form $30°$ with $\overline{OB}$ .