1. **State the problem:** Find the equation of the tangent line to a circle at a given point on the circle in rectangular coordinates. 2. **Recall the circle equation:** A circle with center $(h,k)$ and radius $r$ is given by $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Tangent line property:** The tangent line at point $P(x_1,y_1)$ on the circle is perpendicular to the radius drawn from the center $(h,k)$ to $P$. 4. **Find the slope of the radius:** $$ m_{radius} = \frac{y_1 - k}{x_1 - h} $$ 5. **Find the slope of the tangent line:** Since tangent is perpendicular to radius, $$ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{x_1 - h}{y_1 - k} $$ 6. **Write the tangent line equation:** Using point-slope form at $P(x_1,y_1)$, $$ y - y_1 = m_{tangent}(x - x_1) $$ 7. **Summary:** Given circle center $(h,k)$ and point of tangency $(x_1,y_1)$, the tangent line is $$ y - y_1 = -\frac{x_1 - h}{y_1 - k}(x - x_1) $$ This formula gives the equation of the tangent line in rectangular coordinates passing through the contact point on the circle.