Chapter 9 · Introduction

Chapter 9 — Straight Lines

Roads, ramps, railway tracks and rays of light: describe a straight line by one number — its slope $m=\tan\theta$ — and by one equation, then measure angles between lines and the shortest distance from a point to a line.

In April 1802 a British officer, William Lambton, and his team stretched steel chains across the flat ground near St Thomas Mount in Madras and measured a straight baseline about $12\ \text{km}$ long, correcting every chain length for the day's temperature. From that single line grew the Great Trigonometrical Survey of India: a web of triangles that crept north for half a century, crossed the Deccan and the Gangetic plain and eventually fixed the height of the highest peak on Earth. Every map you have ever used rests on the same idea — fix a few points by their coordinates, join them with straight lines, and compute everything else.

Straight lines are also how engineers talk about steepness. Indian Railways puts up gradient boards reading "1 in 200" — the track rises $1\ \text{m}$ for every $200\ \text{m}$ it runs horizontally — and the old Bhor Ghat incline near Mumbai, one of the steepest main-line climbs in the country, is about "1 in 37". Building guidelines ask for wheelchair ramps no steeper than about $1$ in $12$. Each of these numbers is a slope: rise divided by run. In this chapter you will see that one number, the slope $m=\tan\theta$, together with one point, pins down a line completely; that slopes tell you at a glance whether two lines are parallel or perpendicular and what angle they meet at; that every line has an equation, which can be written in several useful forms; and, finally, how to compute the shortest distance from a point to a line — how close a ship sailing a straight course comes to a lighthouse, or how far a village is from a new straight highway.

Slope $m=\tan\theta=\dfrac{y_2-y_1}{x_2-x_1}$→Parallel, perpendicular, angle between lines→Equations: $y-y_1=m(x-x_1)$ and friends→Distance from a point to a line
🎯 By the end of this chapter you can…
  • find the inclination and the slope of a line, from its angle or from two points on it, and read the sign of the slope from a sketch;
  • decide whether two lines are parallel or perpendicular from their slopes, and use this to test triangles and quadrilaterals for right angles and parallel sides;
  • find the acute angle between two lines from their slopes, and find a slope that makes a given angle with a given line;
  • write the equation of a line in the most convenient form — horizontal or vertical, point-slope, two-point, slope-intercept or intercept form — and move between these forms;
  • find the perpendicular distance of a point from a line and the distance between two parallel lines, and use them in problems on triangles, roads and paths.
Teaser — One Line, Four Questions

A line turns about the fixed point $A(-3,-1)$. For every position the readouts answer the four questions of this chapter: what angle does it make with the $x$-axis, how steep is it (its slope), what is its equation, and how far is it from the point $P(4,3)$? The green segment is the shortest path from $P$ to the line — it always meets the line at a right angle. Press Play, or set the angle yourself.

30°
angle $\theta$—
slope $m=\tan\theta$—
equation—
distance of $P$—

Sections in this chapter

9.1

Slope of a Line and the Angle between Two Lines

A quick recall of the distance and section formulae; inclination and slope; slopes of parallel and perpendicular lines; the angle between two lines.

$m=\dfrac{y_2-y_1}{x_2-x_1},\ \ m_1m_2=-1$
Open section →9.2

Various Forms of the Equation of a Line

Horizontal and vertical lines, the point-slope, two-point, slope-intercept and intercept forms, and choosing the right form for the data you are given.

$y-y_1=m(x-x_1),\ \ \dfrac xa+\dfrac yb=1$
Open section →9.3

Distance of a Point from a Line

The perpendicular distance of a point from a line, the distance between two parallel lines, and the foot of the perpendicular.

$d=\dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$
Open section →
✅ Before you start
From Class 10 Mathematics you need coordinate geometry (Chapter 7, Coordinate Geometry): plotting points, the distance formula and the section and mid-point formulae — Section 9.1 recalls them briefly; linear equations in two variables and solving a pair of them (Chapter 3, Pair of Linear Equations in Two Variables), which is how you find where two lines meet; and the trigonometric ratios of $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$ (Chapter 8, Introduction to Trigonometry). From this course you need the tangent of any angle, including obtuse angles, with $\tan(180^\circ-\theta)=-\tan\theta$ (Section 3.2), and the formula for $\tan(A-B)$ (Section 3.3), which gives the angle between two lines. The modulus $|x|$ from Section 2.3 appears in the angle and distance formulae, because angles and distances are never negative.
← 8.3 Arithmetic and Geometric Means9.1 Slope of a Line and the Angle between Two Lines →