CBSE · Class 10 · Mathematics · Chapter 7

Coordinate Geometry — Formula Sheet

Board Formulas

8 formulas

1.Distance Between Two Points★
$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
$(x_{1}, y_{1})$ : Coordinates of the first point · $(x_{2}, y_{2})$ : Coordinates of the second point · $d$ : Distance (units)
Direct application of Pythagoras in the plane. Distance is always positive.
2.Distance from Origin
$d = x^{2} + y^{2}$
Special case with (x₁, y₁) = (0, 0). Same as the magnitude of the position vector.
3.Section Formula (Internal Division)★
$P = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$
P divides AB in the ratio m:n internally (from A to B). Weights cross-over: m goes with x₂, n with x₁.
4.Midpoint Formula★
$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Section formula with 1:1 ratio. Average the coordinates.
5.Centroid of a Triangle
$G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$
Average the coordinates of the three vertices. G divides each median in the ratio 2:1.
6.Area of a Triangle★
$Area = \frac{1}{2} x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})$
Absolute value ⇒ area is always positive. Zero means the three points are collinear.
7.Collinearity Test
$x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) = 0$
Three points collinear ⇔ triangle area is zero.
8.Trisection Points
$P_{1} = A + \frac{1}{3} (B - A), P_{2} = A + \frac{2}{3} (B - A)$
Two trisection points divide AB in ratios 1:2 and 2:1 respectively. Use section formula twice.