Maharashtra State Board · Class 10 · Mathematics · Geometry Chapter 5

Coordinate Geometry — Formula Sheet

Board Formulas

9 formulas

1.Distance Formula★
$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Length of segment joining (x₁, y₁) and (x₂, y₂). Always positive.
2.Section Formula (Internal Division)★
$(x, y) = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$
$m, n$ : Ratio in which the point divides the segment · $(x_{1}, y_{1})$ : First endpoint · $(x_{2}, y_{2})$ : Second endpoint
Point dividing segment from (x₁,y₁) to (x₂,y₂) in ratio m:n internally.
3.Midpoint Formula
$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Special case of section formula with m = n = 1.
4.Area of Triangle★
$Area = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣$
Take absolute value. If area = 0, three points are collinear.
5.Collinearity Condition
$x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) = 0$
Three points collinear ⇔ area of triangle they form is zero.
6.Centroid of Triangle
$G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$
Divides each median in the ratio 2:1 from vertex to midpoint.
7.Slope Between Two Points
$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Slope tells the inclination. Vertical line: slope undefined; horizontal: slope = 0.
8.Section — External Division
$(x, y) = (\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n})$
Point lies on extended line outside the segment.
9.Distance from Origin
$d = x^{2} + y^{2}$
Special case of distance formula with (x₁, y₁) = (0, 0).