Chapter 3.8, Problem 9PSB

To determine

To prove : The congruency of $\overrightarrow{RK}\text{ and }\overrightarrow{JO}$ .

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{RK}\perp \overrightarrow{HR}$

  $\overrightarrow{JO}\perp \overrightarrow{PM}$

  $\overrightarrow{PH}\cong \overrightarrow{PM}$

  $\overrightarrow{PR}\cong \overrightarrow{PO}$

Formula used : Any two triangles are said to be congruent by ASA property if they have two congruent angles, and the side containing them is also congruent

Proof : We know that

  $\overrightarrow{PH}\cong \overrightarrow{PM}$

  $\overrightarrow{PR}\cong \overrightarrow{PO}$

Thus, we subtract the second from the first. Hence, we get

  $\overrightarrow{RH}\cong \overrightarrow{OM}$

Now, we also know that $\overrightarrow{PH}\cong \overrightarrow{PM}$ . Thus, PHM is an isosceles triangle. Thus,

  $\angle RHK\cong \angle OMJ$

Also,

  $\angle HRK\cong \angle MOJ={90}^0$ (perpendicular lines)

Thus, we can say that

  $\Delta HRK\cong \Delta MOJ$ (by ASA property mentioned above)

Hence, by congruency, $\overrightarrow{RK}\cong \overrightarrow{JO}$