Chapter 8.3, Problem 10CE

To determine

  $$

∠P

To find: the reason of angle is right angle.

Answer to Problem 10CE

The angle $\angle P$ is right angle.

Explanation of Solution

Given:

The triangles are shown below:

Calculation:

Consider triangles $\Delta POQ$ and $\Delta ROS$ ,

  $\begin{array}{l}\angle ROS=\angle POQ\left(\text{Vertical opposite angle}\right)\\ \frac{PO}{RO}=\frac{6}{9}=\frac{2}{3}\left(\text{Given}\right)\\ \frac{QO}{SO}=\frac{10}{15}=\frac{2}{3}\left(\text{Given}\right)\\ \therefore \frac{PO}{RO}=\frac{QO}{SO}\\ \therefore \Delta POQ\sim \Delta ROS\left(SAS\right)\end{array}$

Hence,

  $\begin{array}{l}\frac{PQ}{RS}=\frac{PO}{RO}=\frac{QO}{SO}=\frac{2}{3}\\ \frac{PQ}{12}=\frac{2}{3}\\ PQ=8\end{array}$

Now, take square root of addition of the square to lengths OP and PQ as follows:

  $\begin{array}{l}\Rightarrow \sqrt{O{P}^2+P{Q}^2}\\ \Rightarrow \sqrt{6^2+8^2}\\ \Rightarrow \sqrt{36+64}\\ \Rightarrow \sqrt{100}\\ \Rightarrow 10=OQ\end{array}$

Thus, this proves the Pythagoras theorem.

Hence, angle $\angle P$ is right angle.