Coordinate Geometry

Geometry is the right foundation of all painting. — Albrecht Dürer

7.1 Introduction

Points in the plane are given by coordinates $(x,y)$. This chapter uses algebra to find distances and dividing points on a line segment.

7.2 Distance Formula

Distance between $P(x_1,y_1)$ and $Q(x_2,y_2)$:

$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Distance from origin to $P(x,y)$ is $\sqrt{x^2+y^2}$.

Applications: show collinearity, find side lengths of triangles, verify types of triangles.

7.3 Section Formula

Point dividing the join of $A(x_1,y_1)$ and $B(x_2,y_2)$ in ratio $m:n$ internally:

$\left(\frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n}\right)$

Midpoint (ratio $1:1$):

$\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$

Chapter Summary

• Distance: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
• Section (internal): $\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$
• Midpoint: average of coordinates

Exercises (NCERT)

• Exercise 7.1 — distance formula
• Exercise 7.2 — section formula, midpoint
• Miscellaneous Exercise