Coordinate Geometry | Class 10 Mathematics Notes

7.3 Section Formula

The Section Formula is a key concept in coordinate geometry used to find the coordinates of a point that divides a line segment joining two points in a given ratio.

Consider two points A(x₁, y₁) and B(x₂, y₂) on the coordinate plane. Suppose a point P(x, y) lies on the line segment AB such that it divides AB internally in the ratio m₁ : m₂. This means that the length of segment AP is to the length of segment PB as m₁ is to m₂.

To find the coordinates of P, we use the similarity of triangles formed by dropping perpendiculars from points to the axes and applying the properties of similar triangles (AA similarity criterion). By comparing the ratios of corresponding sides, we derive the formula:

x = (m₁ x₂ + m₂ x₁) / (m₁ + m₂)

y = (m₁ y₂ + m₂ y₁) / (m₁ + m₂)

This formula gives the coordinates of the point P dividing the segment AB internally in the ratio m₁ : m₂.

Special cases include:

• If m₁ = m₂ = 1, then P is the midpoint of AB, and its coordinates are ((x₁ + x₂)/2, (y₁ + y₂)/2).
• If the ratio is k : 1, the formula simplifies accordingly.

The section formula can also be used to find the ratio in which a given point divides a line segment by equating the coordinates and solving for the ratio.

Examples in this section illustrate:

• Finding coordinates of a point dividing a segment in a given ratio.
• Determining the ratio in which a point divides a segment.
• Finding points of trisection (dividing a segment into three equal parts).
• Finding the point where a line segment is divided by an axis.
• Using properties of parallelograms, such as diagonals bisecting each other, to find unknown coordinates.

This section also introduces the idea of external division, where the point dividing the segment lies outside the segment, to be studied in higher classes.

The section formula is a powerful tool for solving many geometric problems algebraically.

📊 Diagram: Fig. 7.9 shows a point P dividing AB in ratio 1:2; Fig. 7.10 illustrates the similar triangles used to derive the section formula; Fig. 7.11 shows points of trisection dividing a segment into three equal parts.

🧪 Activity: Derive the section formula using similar triangles and apply it to find coordinates of points dividing segments in given ratios.

🔗 Connection: Understanding the section formula prepares students to solve more complex problems involving line segments, midpoints, and coordinate calculations, leading to applications in area calculations and further geometric concepts.