Algebra - Coordinate Geometry (Reflection, Section Formula, Equation of a Line)

Reflection in Axes and Origin: Reflection is a transformation where a point is 'flipped' over a line. In the $x$-axis, the point $(x, y)$ becomes $(x, -y)$, appearing as a vertical mirror image across the horizontal axis. In the $y$-axis, $(x, y)$ becomes $(-x, y)$, representing a horizontal flip across the vertical axis. Reflection in the origin $(0, 0)$ transforms $(x, y)$ into $(-x, -y)$, which visually looks like a $180^{\circ}$ rotation around the center.

Invariant Points: A point is considered invariant with respect to a line (or point) of reflection if its coordinates remain unchanged after the transformation. Visually, these are points that lie exactly on the mirror line. For example, any point on the $x$-axis like $(5, 0)$ is invariant when reflected in the $x$-axis.

Section Formula (Internal Division): This formula determines the coordinates of a point $P$ that divides a line segment $AB$ into two parts with a specific ratio $m_1:m_2$. Visually, if $P$ is closer to $A$, $m_1$ is smaller than $m_2$. The formula uses a weighted average of the coordinates of the endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$.

Midpoint and Centroid: The midpoint is a special case of the section formula where the ratio is $1:1$, representing the exact center of a line segment. The centroid of a triangle is the point where its three medians intersect; visually, it is the 'balance point' of the triangle, calculated by averaging the $x$ and $y$ coordinates of the three vertices.

Slope (Gradient) of a Line: The slope $m$ represents the steepness and direction of a line. It is defined as the tangent of the angle of inclination $\theta$ ($m = \tan \theta$). On a graph, a positive slope indicates the line rises from left to right, while a negative slope indicates it falls. A horizontal line has a slope of $0$, and a vertical line has an undefined slope.

Forms of the Equation of a Line: A straight line can be expressed in different forms depending on the given data. The Slope-Intercept form $y = mx + c$ is most common, where $c$ is the $y$-intercept (the point where the line crosses the vertical axis). The Point-Slope form is used when one point and the slope are known, describing how the line radiates from that specific point.

Parallel and Perpendicular Lines: Geometrically, parallel lines have the same steepness and never meet, meaning their slopes are equal ($m_1 = m_2$). Perpendicular lines intersect at a $90^{\circ}$ angle; algebraically, the product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$), which means one slope is the negative reciprocal of the other.