Coordinate Geometry - Section Formula (Internal Division)

The Section Formula is used to find the coordinates of a point $P(x, y)$ that divides a line segment joining two given points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a specific ratio $m_1:m_2$.

Internal Division occurs when the point $P$ lies on the line segment $AB$ between the points $A$ and $B$. Visually, if you imagine a string stretched from $A$ to $B$, point $P$ is a knot tied somewhere on that string, dividing it into two pieces of lengths $m_1$ and $m_2$.

The x-coordinate of the dividing point is calculated as the weighted average of the x-coordinates of the endpoints: $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}$. Similarly, the y-coordinate is $y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$.

The Midpoint is a special case of the section formula where the ratio $m_1:m_2$ is $1:1$. Visually, the point $M$ is located exactly at the center of the segment $AB$, equidistant from both ends.

When calculating the ratio in which a point divides a segment, it is often simpler to assume the ratio is $k:1$. If the resulting value of $k$ is positive, it confirms the division is internal.

Points of Trisection are the two points that divide a line segment into three equal parts. To find them, you apply the section formula twice: once with the ratio $1:2$ and once with the ratio $2:1$.

The Centroid of a triangle is the point where its three medians intersect. Visually, it is the 'balance point' of the triangle. It divides each median in the ratio $2:1$ from the vertex to the midpoint of the opposite side.

Coordinate geometry problems involving the X-axis or Y-axis use the visual property that any point on the X-axis has a y-coordinate of $0$ $(x, 0)$, and any point on the Y-axis has an x-coordinate of $0$ $(0, y)$.