Collinear points, parallel and perpendicular lines

What you will be able to do

• decide from slopes whether two lines are parallel or perpendicular;
• handle horizontal and vertical lines correctly;
• test whether three points are collinear, and find a missing coordinate;
• use slopes to describe a shape such as a parallelogram or a right triangle.

Parallel lines

Parallel lines lean the same way, so they make the same angle with the $x$-axis. Equal angles of inclination mean equal tangents.

Two lines with slopes $m_1$ and $m_2$ are parallel exactly when $m_1 = m_2$.

Two vertical lines are parallel too, but they have no slopes to compare. A line that has the same slope and passes through the same point is not a second line at all: the two lines are the same.

Perpendicular lines

Turn a line through $90^\circ$ and its inclination becomes $\theta_2 = \theta_1 + 90^\circ$. Then

Two lines with slopes $m_1$ and $m_2$ are perpendicular exactly when $m_1 m_2 = -1$.

The same fact comes from the angle formula. When $1 + m_1 m_2 = 0$, the denominator is $0$ and the angle is $90^\circ$.

A line of slope $\dfrac{2}{3}$ is perpendicular to a line of slope $-\dfrac{3}{2}$: flip the fraction and change the sign. A horizontal line, with slope $0$, is perpendicular to a vertical line, which has no slope. The product rule does not apply to that pair, so remember it separately.

Collinear points

Three points $A$, $B$ and $C$ are collinear when they lie on one straight line. Segments $AB$ and $BC$ share the point $B$. So they lie on one line exactly when they have the same slope.

$A$, $B$, $C$ are collinear exactly when the slope of $AB$ equals the slope of $BC$.

If all three points have the same $x$-coordinate, they lie on a vertical line and are collinear. A slope test cannot be used there.

The test is the same as the parallel test, with a shared point that forces the two lines to be one.

Worked examples

A missing coordinate. Find $k$ so that $A(1, 3)$, $B(4, k)$ and $C(7, 9)$ are collinear.

1. The slope of $AC$. $\dfrac{9 - 3}{7 - 1} = 1$.
2. The slope of $AB$. $\dfrac{k - 3}{4 - 1} = 1$, so $k - 3 = 3$.
3. The answer. $k = 6$.

A parallelogram. Show that $P(0, 0)$, $Q(5, 2)$, $R(7, 5)$ and $S(2, 3)$ are the corners of a parallelogram, taken in this order.

1. One pair of sides. $PQ$ has slope $\dfrac{2}{5}$, and $SR$ has slope $\dfrac{5 - 3}{7 - 2} = \dfrac{2}{5}$. They are parallel.
2. The other pair. $QR$ has slope $\dfrac{5 - 2}{7 - 5} = \dfrac{3}{2}$, and $PS$ has slope $\dfrac{3}{2}$. They are parallel.
3. The answer. Both pairs of opposite sides are parallel, so $PQRS$ is a parallelogram. It is not a rectangle, because $\dfrac{2}{5} \times \dfrac{3}{2} = \dfrac{3}{5}$ is not $-1$.

Practise with new numbers

Try it in Python

The program tests collinearity by cross-multiplication, so it needs no division and works for vertical lines. It then checks the test on a thousand random triples.

Common mistakes

• Saying that perpendicular lines have opposite slopes. The product is $-1$. Slopes $3$ and $-3$ are not perpendicular.
• Using the product rule on a vertical line. A vertical line has no slope, so use the rule for horizontal and vertical lines.
• Taking equal slopes to mean the same line. Equal slopes give parallel lines, and they may be the same line.
• Testing $AB$ against $CD$. Collinearity needs two segments that share a point, such as $AB$ and $BC$.

Remember these

• Parallel lines: $m_1 = m_2$. Perpendicular lines: $m_1 m_2 = -1$.
• Horizontal and vertical lines are perpendicular to each other.
• $A$, $B$, $C$ are collinear when the slope of $AB$ equals the slope of $BC$.
• Cross-multiplying avoids division and works for vertical lines.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the slope of a line perpendicular to a line of slope $\dfrac{5}{4}$?

2. Are the lines through $(0, 1)$ and $(2, 5)$, and through $(1, 0)$ and $(4, 6)$, parallel?

3. Are $(2, 1)$, $(4, 5)$ and $(7, 11)$ collinear?

4. For which $k$ is the line through $(1, 2)$ and $(3, k)$ perpendicular to a line of slope $-1$?

5. Is the line through $(3, 1)$ and $(3, 8)$ parallel to the $y$-axis?