Chapter 7: Coordinate Geometry

Descartes' great idea was to label every point in the plane by a pair of numbers $(x, y)$ . Geometry then becomes algebra: distances, midpoints, intersections , all computable from coordinates. This chapter delivers the two formulas you'll use forever after: the distance formula and the section formula, with the midpoint formula as a corollary.

For board exams: $1$ -mark MCQs on distance/midpoint, $2$ - and $3$ -mark questions on section formula, collinearity tests, and a $4$ - to $5$ -mark problem about identifying a shape (parallelogram, square, rhombus) from given vertices, or finding a point that divides a line segment in a given ratio.

Coordinate geometry is the gateway to higher mathematics. The same ideas , distance, midpoint, ratio , extend to three dimensions, to vectors, to complex numbers, to fields most students meet in calculus. Master them now.

What's inside

Coordinate plane and axes , quick refresher.
Distance formula , derivation via Pythagoras.
Section formula , internal division in a given ratio, with midpoint as a special case.
Collinearity and area , testing whether three points are collinear.
Applications and shape recognition , quadrilaterals from vertices.

Key results / Formula card

Result	Formula
Distance	$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Section formula (internal, ratio $m : n$ )	$\left(\dfrac{m x_2 + n x_1}{m + n},\ \dfrac{m y_2 + n y_1}{m + n}\right)$
Midpoint	$\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)$
Collinearity (area = 0)	$x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0$
Distance from origin	$\sqrt{x^2 + y^2}$

Chapter 7: Coordinate Geometry

What's inside

Key results / Formula card

Sub-topics

Test Your Knowledge

AI Summary