Coordinate Geometry in One Shot Class 10 Maths Chapter 7 Concepts CBSE Class 10 Midterm Exams

Coordinate Geometry: Distance, Section Formula, and Area of Triangles

Coordinate geometry (also called analytic geometry) is the branch of mathematics that uses algebraic methods to solve geometric problems by placing figures on a coordinate plane. René Descartes pioneered this approach in the 17th century, and it remains one of the most powerful tools in mathematics. In CBSE Class 10, Chapter 7 focuses on three key techniques: finding the distance between two points, dividing a line segment in a given ratio, and calculating the area of a triangle using coordinates. These tools connect algebra with geometry and are essential for solving a wide range of problems in mathematics, physics, engineering, and computer graphics.

The distance formula is derived from the Pythagorean theorem. Given two points A(x₁, y₁) and B(x₂, y₂) on the Cartesian plane, the distance AB = √[(x₂ − x₁)² + (y₂ − y₁)²]. This formula allows you to calculate the length of any line segment joining two known points, determine whether three points form an isosceles, equilateral, or scalene triangle, verify if points are collinear (distance sum equals the longest segment), and identify the type of quadrilateral formed by four given points. The section formula finds the coordinates of a point that divides the line segment joining two given points in a specific ratio. If P(x, y) divides the line joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then x = (mx₂ + nx₁)/(m + n) and y = (my₂ + ny₁)/(m + n). A particularly important special case is the midpoint formula, where m = n = 1, giving x = (x₁ + x₂)/2 and y = (y₁ + y₂)/2. The section formula is used extensively to find centroids of triangles, locate points that trisect or bisect line segments, and solve problems involving ratios in geometric figures.

The area of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by the formula: Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If this area comes out to be zero, it means the three points are collinear — they lie on the same straight line. This provides an algebraic test for collinearity that is often faster than checking distances. The formula can also be extended to find the area of quadrilaterals by dividing them into two triangles along a diagonal. Board examination problems frequently ask students to verify whether a given set of points forms a particular type of triangle (right-angled, isosceles, equilateral), to find an unknown coordinate given geometric conditions (such as the point being equidistant from two other points or lying on a circle), or to calculate the area of a polygon whose vertices are given. Mastering these three formulas — distance, section, and area — equips students with a complete toolkit for solving coordinate geometry problems efficiently.

• Distance formula: AB = √[(x₂ − x₁)² + (y₂ − y₁)²] — derived from the Pythagorean theorem.
• Section formula: P divides AB in ratio m:n → x = (mx₂ + nx₁)/(m + n), y = (my₂ + ny₁)/(m + n).
• Midpoint formula (special case m = n): x = (x₁ + x₂)/2, y = (y₁ + y₂)/2.
• Area of triangle: ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; area = 0 means collinear points.
• Use distance formula to classify triangles and quadrilaterals; use section formula for centroids and ratios.