Question

Prove that the point D which is as far behind the mirror as the object A is in front of the mirror lies on $\overleftrightarrow{BC}$. (Hint: Show that $\angle CBE$ and $\angle EBD$ are supplementary.)

Short answer

The proof is shown in the below steps.

Step-by-step solution

Step 1. Given Information.

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

Step 2. 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

$\begin{array}{l}AO\cong OD\\ \angle AOB\cong \angle DOB\\ \overline{BO}\cong \overline{BO}\end{array}$

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

Since EO is a straight line

Hence,

$\begin{array}{l}m\angle ABO=m\angle CBE\left[\text{usethelawofreflection}\right]\\ m\angle EBD+m\angle OBD=180°\left[\text{EOisastraightline}\right]\\ m\angle EBD+m\angle ABO=180°\left[m\angle ABO=m\angle OBD\right]\\ m\angle EBD+m\angle CBE=180°\left[m\angle ABO=m\angle CBE\right]\end{array}$

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

Step 3. Conclusion.

The given condition is proved.