Chapter 1.5, Problem 22PSC

To determine

To Calculate: The measure of angle ROS.

Answer to Problem 22PSC

  $m\angle ROS=10°$

Explanation of Solution

Given:

  

  $\begin{array}{l}\overrightarrow{OP}\text{ and }\overrightarrow{OP}\text{ trisect }\angle NOS\\ m\angle NOP=3x-4y\\ m\angle POR=x-y\\ m\angle ROS=y-10\end{array}$

Concept Used:

Trisected line divides the angle in three equal part.

Calculation:

Here, we have

  $\begin{array}{l}m\angle NOP=3x-4y\\ m\angle POR=x-y\\ m\angle ROS=y-10\\ \overrightarrow{OP}\text{ and }\overrightarrow{OP}\text{ trisect }\angle NOS\\ \Rightarrow \angle NOP=\angle POR=\angle ROS\\ \Rightarrow 3x-4y=x-y=y-10\\ \because 3x-4y=y-10\\ \Rightarrow 3x-5y=-10\left(1\right)\\ \because x-y=y-10\\ \Rightarrow x=2y-10\left(2\right)\\ \text{Using }\left(2\right)\text{in }\left(1\right)\text{, we get}\\ 3\left(2y-10\right)-5y=-10\\ \Rightarrow 6y-30-5y=-10\\ \Rightarrow y=30-10\\ \Rightarrow y=20\left(3\right)\\ \text{Using }\left(3\right)\text{in }\left(2\right)\text{, we get}\\ x=2\times 20-10=10\\ \Rightarrow m\angle ROS=\left(y-10\right)°=\left(20-10\right)°=10°\\ \Rightarrow m\angle ROS=10°\end{array}$

Conclusion:

  $m\angle NOP=m\angle POR=m\angle ROS=10°$