Brief Recall of 2D Coordinate Geometry

Before developing the geometry of the straight line, we briefly recall the coordinate-geometry tools built in earlier classes that this chapter uses directly.

Cartesian coordinates. Two mutually perpendicular number lines — a horizontal x-axis and a vertical y-axis — meeting at the origin $O(0,0)$ divide the plane into four quadrants. Every point $P$ in the plane is located by an ordered pair $(x,y)$: $x$ is the signed perpendicular distance of $P$ from the y-axis, and $y$ is its signed perpendicular distance from the x-axis.

Distance formula (recall). For two points $A(x_1,y_1)$ and $B(x_2,y_2)$, drop perpendiculars to form a right triangle with legs $|x_2-x_1|$ and $|y_2-y_1|$; by Pythagoras' theorem,

This is the tool used repeatedly below to measure lengths between named points.

Section formula (recall). The point $P(x,y)$ that divides the segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ internally in the ratio $m:n$ is

Taking $m=n=1$ gives the midpoint formula as a special case: the midpoint of $AB$ is $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$.

What this chapter develops. Having recalled how to locate points and measure the distance between two of them, this chapter turns to the straight line itself — the set of every point satisfying one fixed linear relationship between $x$ and $y$. Section 2 defines the slope of a line, the single number capturing its steepness and direction. Section 3 uses slope to find the angle between two intersecting lines, and reads off from it the special conditions for two lines to be parallel or perpendicular. Sections 4–8 build up the equation of a line in every standard form met at this level — lines parallel to an axis, the point-slope form, the slope-intercept form, the two-point form, and the intercept form — each derived from the one before it by a short chain of algebra, never invented afresh. Section 9 closes the chapter by finding the perpendicular distance of a point from a line, and, as a direct corollary, the distance between two parallel lines.