Coordinate Geometry — Cheat Sheet

Class X · Mathematics Ch.7 · 8 Panels · Quick Reference

📍 The Cartesian Plane
A point = (x, y)
x = abscissa (from y-axis)  |  y = ordinate (from x-axis)
Origin O = (0, 0)
Sign of Coordinates by Quadrant
Quadrantxy
I++
II−+
III−−
IV+−
On the axes: x-axis → (x, 0); y-axis → (0, y).
📏 Distance Formula
Between P(x₁,y₁) and Q(x₂,y₂):
PQ = √[(x₂−x₁)² + (y₂−y₁)²]
Distance from origin:
OP = √(x² + y²)
Proving Figures
FigureCondition
Isosceles Δtwo sides equal
Equilateral Δall three sides equal
Squareall sides equal & diagonals equal
Rhombusall sides equal, diagonals unequal
Careful: always square the differences before adding. Never forget the square root at the end.
✂️ Section Formula
P divides A(x₁,y₁)–B(x₂,y₂) internally in m : n:
P = ( (mx₂+nx₁)/(m+n) , (my₂+ny₁)/(m+n) )
Memory aid: the ratio part touching a point uses the OTHER point's coordinate — m goes with x₂, n goes with x₁.
Finding the Ratio (k method)
  1. Let the ratio be k : 1
  2. Write x = (kx₂ + x₁)/(k+1)
  3. Solve for k, then the ratio is k : 1
Trisection: points dividing AB in 1:2 and 2:1 split it into three equal parts.
⚖️ Midpoint Formula
Midpoint of A(x₁,y₁) and B(x₂,y₂):
M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
Just the average of the x's and the average of the y's — the section formula with ratio 1:1.
Parallelogram trick: diagonals bisect each other, so midpoint of one diagonal = midpoint of the other. Equate them to find a missing vertex.
Centroid of a triangle:
G = ( (x₁+x₂+x₃)/3 , (y₁+y₂+y₃)/3 )
🔺 Area of a Triangle
Vertices (x₁,y₁), (x₂,y₂), (x₃,y₃):
Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Always use | | (modulus) so the area is never negative.
Collinearity Test
Three points are collinear ⇔ Area = 0
i.e. x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂) = 0
Quadrilateral area: split into two triangles with a diagonal, then add their areas.
⭐ Key Results
Centroid divides each median in the ratio 2 : 1 (measured from the vertex).
Distance from an axis: point (x, y) is |y| from the x-axis and |x| from the y-axis.
Equidistant point: if P is equidistant from A and B, set PA² = PB² and simplify — the squares remove the roots.
Point on an axis: a point on the x-axis is (x, 0); on the y-axis is (0, y). Use this to reduce unknowns.
🌍 Common Applications
Classify QuadrilateralFind all sides & diagonals; compare to identify square/rhombus/rectangle/parallelogram
Find Missing VertexUse midpoint/section/area equations to solve for unknown coordinates
Seating / MapsLocate points and find distances on a coordinate grid
Dividing a PathSection formula finds where a point splits a route in a given ratio
Approach: plot roughly, choose the right formula (distance / section / area), substitute, and simplify carefully.
🧠 Exam Quick Recall
QuantityFormula
Distance√[(x₂−x₁)²+(y₂−y₁)²]
Section (m:n)((mx₂+nx₁)/(m+n), …)
Midpoint((x₁+x₂)/2, (y₁+y₂)/2)
Centroid((Σx)/3, (Σy)/3)
Area½|x₁(y₂−y₃)+…|
CollinearArea = 0
Common mistakes: mixing up (x,y) order; forgetting the modulus in area; swapping m and n in the section formula.