Chapter Summary: Coordinate Geometry

A one-page recap of every key idea and formula in this chapter. Read this the night before the exam.

1. The Cartesian Plane

  • A point is an ordered pair (x,y)(x, y): xx is the abscissa, yy is the ordinate.
  • Quadrant signs (anticlockwise): I (+,+)(+,+), II (−,+)(-,+), III (−,−)(-,-), IV (+,−)(+,-).
  • On the x-axis y=0y = 0; on the y-axis x=0x = 0; origin is (0,0)(0, 0).

2. Distance Formula

AB=(x2−x1)2+(y2−y1)2,OP=x2+y2.AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, \qquad OP = \sqrt{x^2 + y^2}.

  • Use it to test shapes (equilateral, isosceles, right triangle, square, rhombus, etc.) and collinearity (one distance = sum of the other two).

3. Section and Midpoint Formulas

  • Section formula (internal ratio m:nm:n): (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).
  • Midpoint (1:11:1): (x1+x22,y1+y22)\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right).
  • To find the ratio, set it to k:1k:1 and solve; positive kk means internal division.
  • Trisection points use ratios 1:21:2 and 2:12:1.
  • A missing vertex of a parallelogram is found by equating diagonal midpoints.

4. Area of a Triangle and Collinearity

Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣.\text{Area} = \tfrac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.

  • Three points are collinear   ⟺  \iff this area is 0.
  • Quadrilateral area = sum of two triangle areas via a diagonal. (Area-of-triangle is retained by many State Boards.)

5. The Centroid

G=(x1+x2+x33,  y1+y2+y33).G = \left(\frac{x_1+x_2+x_3}{3},\; \frac{y_1+y_2+y_3}{3}\right).

  • The centroid is the average of the three vertices and divides each median 2:12:1 from the vertex.

Common mistakes to avoid

  • Forgetting the modulus or the 12\tfrac12 in the area formula.
  • Dividing by 2 instead of 3 when finding the centroid.
  • Swapping the section-formula pattern — remember mm pairs with x2x_2, nn with x1x_1.
  • Mixing up 'equal sides' (rhombus) with 'equal sides + equal diagonals' (square).

Final tip: Decide which of the four tools the question needs — distance, section/midpoint, area, or centroid — write the formula, then substitute carefully. That single habit secures most of the marks in this chapter.