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Class 10 · Chapter 7 · Mathematics

Coordinate Geometry Cheat Sheet

Formulae

Distance formula

PQ=(x2−x1)2+(y2−y1)2PQ = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}

The distance between the points P(x₁, y₁) and Q(x₂, y₂).

Distance from the origin

OP=x2+y2OP = \sqrt{x^{2} + y^{2}}

The distance of the point P(x, y) from the origin O(0, 0).

Section formula (internal division)

(mx2+nx1m+n,my2+ny1m+n)\left(\frac{m x_{2} + n x_{1}}{m+n}, \frac{m y_{2} + n y_{1}}{m+n}\right)

Coordinates of the point dividing the segment joining (x₁, y₁) and (x₂, y₂) in the ratio m : n internally.

Midpoint formula

(x1+x22,y1+y22)\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂); the section formula with m = n.

Section formula (external division)

(mx2−nx1m−n,my2−ny1m−n)\left(\frac{m x_{2} - n x_{1}}{m-n}, \frac{m y_{2} - n y_{1}}{m-n}\right)

Coordinates of the point dividing the segment externally in the ratio m : n.

Area of a triangle

A=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣A = \frac{1}{2}\left|x_{1}(y_{2}-y_{3}) + x_{2}(y_{3}-y_{1}) + x_{3}(y_{1}-y_{2})\right|

For a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃).

Definitions

Cartesian plane
The plane formed by two perpendicular number lines, the x-axis and the y-axis.
Origin
The point (0, 0) where the x-axis and y-axis meet.
Abscissa
The x-coordinate of a point — its distance from the y-axis.
Ordinate
The y-coordinate of a point — its distance from the x-axis.
Quadrants
The four regions into which the axes divide the plane, numbered I to IV anticlockwise.
Collinear points
Points that lie on the same straight line.

Key Points

  • The distance formula is derived from the Pythagoras theorem.
  • On the x-axis the y-coordinate is 0; on the y-axis the x-coordinate is 0.
  • A point (x, y) lies in the first quadrant if x > 0 and y > 0, and so on for the other quadrants.
  • The distance of a point from the origin is √(x² + y²).
  • The midpoint of a segment is obtained by the section formula with equal division ratio 1 : 1.
  • Three points are collinear if the area of the triangle formed by them is zero.
  • The section formula gives the coordinates of a point dividing a segment in a given ratio.
  • Distance is always non-negative; the distance of a point from itself is zero.