Given a point $M(x ; y)$. Find the coordinates of the point symmetric to point $M$ with respect to: a) the OX axis; b) the OY axis.

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Let the point $M^{\prime}\left(x^{\prime} ; y\right)$ be symmetric to the point $M(x ; y)$ with respect to the axis $O X$. Then the points $M$ and $M^{\prime}$ lie on opposite sides of the axis $O X$ on a line perpendicular to this axis, at equal distances from the point $P$ where this line intersects the axis $O X$. Therefore, the points $M$ and $M^{\prime}$ have the same abscissas and opposite ordinates. Hence, $x^{\prime}=x, y^{\prime}=-y$.

For the point $M^{\prime}\left(x^{\prime} ; y^{\prime}\right)$, symmetric to the given point $M(x ; y)$ with respect to the axis $O Y$, we similarly obtain that $x^{\prime}=-x, y^{\prime}=y$.

## Answer

a) $(x ;-y) ;$ b) $(-x ; y)$.