Points, distances and lines in the plane

Points and distance

A point in the plane is a pair $(x, y)$. The distance between $A(x_1, y_1)$ and $B(x_2, y_2)$ is the hypotenuse of a right triangle whose legs are the horizontal gap $x_2 - x_1$ and the vertical gap $y_2 - y_1$:

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

Because each gap is squared, the order of the points does not matter and negative gaps cause no harm. The midpoint of $AB$ averages the coordinates:

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

Distance subtracts; midpoint adds. Mixing the two is the commonest slip in this topic.

Slope

The slope of the line through $A$ and $B$ is the vertical change per unit of horizontal change:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Subtract in the same order on top and bottom. A positive slope rises from left to right, a negative slope falls, a slope of 0 is a horizontal line, and a vertical line (where $x_2 = x_1$) has no slope at all. The slope is also the tangent of the angle the line makes with the positive $x$-axis, which links this lesson to trigonometry: a slope of 1 means a 45° line.

Equations of a line

Every non-vertical line can be written as $y = mx + q$, where $m$ is the slope and $q$ the $y$-intercept, the height at which it crosses the $y$-axis. Two other forms are just as useful:

• Point and slope. The line through $(x_0, y_0)$ with slope $m$ is $y - y_0 = m(x - x_0)$. This is the fastest way to write a line when a question gives a point and a direction.
• General form. $ax + by + c = 0$ covers every line, vertical ones included. If $b \neq 0$, solving for $y$ gives $y = -\frac{a}{b}x - \frac{c}{b}$, so the slope is $-\frac{a}{b}$.

Vertical lines are $x = h$; horizontal lines are $y = k$. A point lies on a line exactly when its coordinates satisfy the equation, which is also the quickest way to test an answer option.

Parallel and perpendicular

Two non-vertical lines are parallel when their slopes are equal, $m_1 = m_2$. They are perpendicular when the product of their slopes is $-1$, that is $m_2 = -\frac{1}{m_1}$: flip the fraction and change its sign. A line of slope $\frac{2}{3}$ is perpendicular to every line of slope $-\frac{3}{2}$. The special case: a horizontal line is perpendicular to a vertical one, though neither formula covers it.

In general form there is a shortcut. The lines parallel to $ax + by + c = 0$ are $ax + by + k = 0$ for any $k$: keep $a$ and $b$, change the constant. The lines perpendicular to it are $bx - ay + k = 0$: swap $a$ and $b$ and change one sign. Then fix $k$ by substituting the given point.

Where two lines meet

The intersection point satisfies both equations, so solve them as a system, by substitution or by elimination. The system has exactly one solution when the slopes differ, none when the lines are parallel and distinct, and infinitely many when the two equations describe the same line (one is a multiple of the other).

Distance from a point to a line

The shortest distance from $P(x_0, y_0)$ to the line $ax + by + c = 0$ is measured along the perpendicular:

$$d = \frac{\lvert ax_0 + by_0 + c \rvert}{\sqrt{a^2 + b^2}}$$

The line must be in general form, with everything on one side, before you read off $a$, $b$ and $c$. This formula returns in the lesson on lines and conics, where comparing a distance with a radius decides whether a line cuts a circle.

Working without a calculator

Coordinates in these questions are chosen to give tidy numbers: gaps of 6 and 8 give a distance of 10, a denominator of $\sqrt{3^2 + 4^2} = 5$ divides cleanly. If your working produces an ugly root, recheck a subtraction before you trust it.