Solution:

Suppose $A$ be the set of all points in a plane and $O$ be the origin. (As given)
Therefore, $\mathrm{R}=\{(\mathrm{P}, \mathrm{Q}): \mathrm{P}, \mathrm{Q} \in \mathrm{A}$ and $\mathrm{OP}=\mathrm{OQ})\}$
If $R$ is Reflexive, Symmetric and Transitive, then $R$ is an equivalence relation.
So now,
Reflexivity:
For any point $\mathrm{P}$, we have
$\begin{array}{l}
\mathrm{OP}=\mathrm{OP} \\
\Rightarrow(\mathrm{P}, \mathrm{P}) \in \mathrm{R}
\end{array}$
Therefore, $(\mathrm{P}, \mathrm{P}) \in \mathrm{R}$ for all $\mathrm{P} \in \mathrm{A}$
As a result, $R$ is reflexive.
Symmetric:
Let $\mathrm{P}$ and $\mathrm{Q}$ inset $\mathrm{A}$ such that,
$(\mathrm{P}, \mathrm{Q}) \in \mathrm{R}$
$\Rightarrow \mathrm{OP}=\mathrm{OQ}$
$\begin{array}{l}
\Rightarrow \mathrm{OQ}=\mathrm{OP} \\
\Rightarrow(\mathrm{Q}, \mathrm{P}) \in \mathrm{R}
\end{array}$
Therefore, $(\mathrm{P}, \mathrm{Q}) \in \mathrm{R} \Rightarrow(\mathrm{Q}, \mathrm{P}) \in \mathrm{R}$ and $(\mathrm{P}, \mathrm{Q}) \in \mathrm{A}$
As a result, $\mathrm{R}$ is symmetric.
Transitivity:
Suppose $\mathrm{P}, \mathrm{Q}$ and $\mathrm{S}$ are three points in set $\mathrm{A}$, such that $(\mathrm{P}, \mathrm{Q}) \in \mathrm{R}$ and
$(\mathrm{Q}, \mathrm{S}) \in \mathrm{R}$
$\begin{array}{l}
\Rightarrow \mathrm{OP}=\mathrm{OQ} \text { and } \mathrm{OQ}=\mathrm{OS} \\
\Rightarrow \mathrm{OP}=\mathrm{OS} \\
\Rightarrow(\mathrm{P}, \mathrm{S}) \in \mathrm{R}
\end{array}$
Therefore, $R$ is transitive.
As a result, $R$ is an equivalence relation.