Chapter 14.2, Problem 5E

To determine

To prove: that the point D which is as far behind the mirror as the object A is in front of the mirror lies on $\overleftrightarrow{BC}$ .

Explanation of Solution

Given information:

The given statement is that the condition can be proved by proving that $\angle CBE$ and $\angle EBD$ are supplementary.

Proof:

The geometric diagram to proof the condition.

To show that the point D lies on $\overleftrightarrow{BC}$ i.e. $\angle CBE+\angle EBD=180°$

In triangle AOB and DOB

$\begin{array}{l}AO\cong OD\\ \angle AOB\cong \angle DOB\\ \overline{BO}\cong \overline{BO}\\ \text{triangle }AOB\cong \text{ triangle }DOB\left[\text{side angle side}\right]\\ \text{hence, }m\angle ABO=m\angle OBD\end{array}$

Since EO is a straight line

Hence,

  $\begin{array}{l}m\angle ABO=m\angle CBE\left[\text{use the law of reflection}\right]\\ m\angle EBD+m\angle OBD=180°\left[\text{EO is a straight line}\right]\\ m\angle EBD+m\angle ABO=180°\left[m\angle ABO=m\angle OBD\right]\\ m\angle EBD+m\angle CBE=180°\left[m\angle ABO=m\angle CBE\right]\end{array}$

Hence, $\angle CBE$ and $\angle EBD$ are supplementary.

The given condition is proved.