Parallel and Perpendicular Lines, SAT math lesson

Train rails never meet, and the corner of a sheet of paper is a right angle. In the $xy$ -plane both relationships show up in the slopes. Parallel lines climb at the same rate, and perpendicular lines turn a rise into a run. Once you see that, a question asking for the line through a point parallel or perpendicular to another line becomes two short steps.

See also Drawing Lines from Slope and Intercepts in Foundations for Calculus.

Key terms

Parallel lines

Definition Two distinct lines in a plane are parallel when they have no point in common. Two nonvertical lines $y = m_1x + b_1$ and $y = m_2x + b_2$ are parallel exactly when

m_1 = m_2 \quad \text{and} \quad b_1 \ne b_2.

Any two distinct vertical lines $x = a_1$ and $x = a_2$ are also parallel. A vertical line is never parallel to a nonvertical one, since the two always cross.

In plain words Same steepness, different starting height. The lines run side by side forever.

Example Find the line through $(2, 1)$ parallel to $y = 3x - 4$ . It needs slope $3$ , so $y - 1 = 3(x - 2)$ , which gives $y = 3x - 5$ . Its $y$ -intercept $-5$ differs from $-4$ , so the two lines really are distinct and parallel.

Parallel or the same line

Definition Two equations can describe one and the same line. Two lines coincide when every point of one is a point of the other. For nonvertical lines this happens exactly when the slopes and the $y$ -intercepts are both equal. On this page, including the questions at the end, parallel always means two distinct lines with no common point, so a line is not called parallel to itself. Some books count a line as parallel to itself, and under that convention equal slopes alone are enough, so read how a question uses the word. Any two lines in a plane are parallel, coincident, or meet in exactly one point, and only one of the three happens.

In plain words Equal slopes alone are not enough. Also compare the intercepts, because the second line might just be the first one written differently.

Example Solving $2x + 3y = 6$ for $y$ gives $y = -\frac{2}{3}x + 2$ . Solving $4x + 6y = 12$ gives the same thing, so these are one line. The equation $4x + 6y = 18$ gives $y = -\frac{2}{3}x + 3$ , with the same slope and a different intercept, so that line is parallel to the first.

Perpendicular lines

Definition Two lines are perpendicular when they meet at a right angle. If both lines are nonvertical, with slopes $m_1$ and $m_2$ , then they are perpendicular exactly when

m_1 \cdot m_2 = -1.

This product rule applies only when both slopes are defined. It also forces both slopes to be nonzero. The remaining case is a horizontal line and a vertical line, which are always perpendicular even though the product of their slopes cannot be formed.

In plain words For two slanted lines, multiply the slopes. If you get $-1$ , they meet at a right angle.

Example Find the line through $(4, 3)$ perpendicular to $y = 2x + 1$ . The new slope must satisfy $2 \cdot m = -1$ , so $m = -\frac{1}{2}$ . Then

y - 3 = -\frac{1}{2}(x - 4),

which simplifies to $y = -\frac{1}{2}x + 5$ . The point checks, since $-2 + 5 = 3$ .

Negative reciprocal

Definition The negative reciprocal of a nonzero real number $m$ is

-\frac{1}{m}.

The number $0$ has no negative reciprocal. A nonvertical line with slope $m \ne 0$ is perpendicular to exactly the lines of slope $-\frac{1}{m}$ . Here is why. A line with slope $\frac{b}{a}$ , where $a$ and $b$ are nonzero, moves $a$ across and $b$ up. Turning that step through a right angle gives $b$ to the left and $a$ up, which is a slope of

\frac{a}{-b} = -\frac{a}{b}.

In plain words Flip the fraction and change its sign. Rotating the slope triangle swaps rise and run, and one of them turns negative.

Example The slope triangle with run $3$ and rise $2$ has slope $\frac{2}{3}$ . Rotated a quarter turn counterclockwise it has run $-2$ and rise $3$ , a slope of $-\frac{3}{2}$ . A few more pairs follow.

Slope	Negative reciprocal	Product
$\frac{3}{4}$	$-\frac{4}{3}$	$-1$
$-5$	$\frac{1}{5}$	$-1$
$-\frac{2}{7}$	$\frac{7}{2}$	$-1$

A common slip is to flip without changing the sign. The slopes $\frac{3}{4}$ and $\frac{4}{3}$ have product $1$ , not $-1$ .

Horizontal lines

Definition A horizontal line is the set of all points with one fixed $y$ -coordinate $k$ , with the equation

y = k.

Its slope is $0$ . Every horizontal line is parallel to every other horizontal line and perpendicular to every vertical line.

In plain words A flat line. It goes neither up nor down, so any line at a right angle to it stands straight up.

Example The line through $(-2, 5)$ parallel to the $x$ -axis is $y = 5$ . The line through $(-2, 5)$ perpendicular to $x = 7$ is the same line $y = 5$ , because a line perpendicular to a vertical line must be horizontal.

Vertical lines

Definition A vertical line is the set of all points with one fixed $x$ -coordinate $h$ , with the equation

x = h.

Its slope is undefined. Every vertical line is parallel to every other vertical line and perpendicular to every horizontal line. Since its slope is not a number, the product rule and the negative reciprocal do not apply to it.

In plain words A line that stands straight up. You cannot take the negative reciprocal of an undefined slope, so handle these lines by picture, not by formula.

Example Find the line through $(6, 1)$ perpendicular to $y = -4$ . The line $y = -4$ is horizontal, with slope $0$ , and $0$ has no negative reciprocal. A perpendicular line must be vertical, so the answer is $x = 6$ . Writing $y = 1$ would give a line parallel to $y = -4$ , not perpendicular to it.

Check yourself

Line $\ell$ is defined by $4x - 3y = 6$ . Line $k$ is perpendicular to $\ell$ and passes through $(8, 1)$ . Which equation defines line $k$ ? (A) $y = -\frac{3}{4}x + 7$ (B) $y = \frac{3}{4}x - 5$ (C) $y = -\frac{4}{3}x + \frac{35}{3}$ (D) $y = -\frac{3}{4}x - 5$

Show the answer

The answer is (A). Solving $4x - 3y = 6$ for $y$ gives slope $\frac{4}{3}$ , so line $k$ has slope $-\frac{3}{4}$ . Then

y - 1 = -\frac{3}{4}(x - 8)

gives $y = -\frac{3}{4}x + 7$ . Choice (B) flips the fraction but keeps the sign, and choice (C) changes the sign but does not flip. Both pass through $(8, 1)$ with the wrong slope. Choice (D) has the right slope but comes from writing $x + 8$ instead of $x - 8$ , and it gives $-11$ at $x = 8$ .

In the $xy$ -plane, the line through $(2, k)$ and $(6, 9)$ is parallel to the line $y = \frac{1}{2}x - 4$ . What is the value of $k$ ?

Show the answer

The answer is $7$ . Parallel lines have equal slopes, so

\frac{9 - k}{6 - 2} = \frac{1}{2},

which gives $9 - k = 2$ and $k = 7$ . The line through $(2, 7)$ and $(6, 9)$ is $y = \frac{1}{2}x + 6$ , whose intercept $6$ differs from $-4$ , so the lines are distinct.

Line $m$ is perpendicular to the line $x = -2$ and passes through $(5, -3)$ . Which equation defines line $m$ ? (A) $x = 5$ (B) $y = -3$ (C) $y = \frac{1}{2}x - \frac{11}{2}$ (D) $x = -3$

Show the answer

The answer is (B). The line $x = -2$ is vertical, so a perpendicular line is horizontal, and the horizontal line through $(5, -3)$ is $y = -3$ . Choice (A) is vertical, so it is parallel to $x = -2$ . Choice (C) comes from reading $x = -2$ as a slope of $-2$ and taking its negative reciprocal $\frac{1}{2}$ . Choice (D) uses the $y$ -coordinate $-3$ in a vertical line, which does not even pass through the point.

Parallel and Perpendicular Lines

Theory, part 27

Practice, part 28

Key terms

Parallel lines

Parallel or the same line

Perpendicular lines

Negative reciprocal

Horizontal lines

Vertical lines

Check yourself

How this lesson connects

Builds on