Lesson 36: Using Parallel and Perpendicular Lines

New Concept

For parallel lines: $m_1 = m_2$. For perpendicular lines: $m_1 m_2 = -1$.

Why it matters

Algebra is the language that translates geometric relationships, like parallel and perpendicular lines, into precise numerical rules you can manipulate. This bridge between visual shapes and symbolic equations is fundamental to everything from video game design to engineering complex structures.

What’s next

Next, you'll use these slope rules to write equations for new lines and prove geometric properties, like whether a triangle is a right triangle.

Parallel lines have identical slopes and different $y$-intercepts. For two parallel lines, their slopes are equal:

To check if lines are parallel, compare their slopes. The line through $(1,1)$ and $(2,4)$ has $m_1 = 3$. The line through $(-1,-1)$ and $(1,5)$ has $m_2 = 3$. Since $m_1=m_2$, they are parallel.
The lines $y = 2x + 1$ and $y = 2x - 100$ are parallel because their slopes are both $2$.

Imagine two skateboards rolling down a hill on perfectly separate paths—they'll never crash! That's because they're parallel, always heading in the same direction with the exact same steepness, or slope. If they also started at the same spot (y-intercept), they would just be one skateboard on top of another, which isn't very parallel, is it?

Perpendicular lines have slopes that are negative reciprocals of each other. The product of the slopes of two perpendicular lines is $-1$.

A line with slope $m_1 = 4$ is perpendicular to a line with slope $m_2 = -\frac{1}{4}$, because $4(-\frac{1}{4}) = -1$.
To find the perpendicular slope for $m = -\frac{2}{3}$, flip the fraction and change the sign to get $m = \frac{3}{2}$.

Perpendicular lines are like the corners of a perfect square, meeting at a crisp 90-degree angle. Their slopes are total opposites! To get one from the other, you flip the fraction (that's the reciprocal part) and switch its sign. When you multiply these opposite-reciprocal slopes together, they always cancel out to make exactly $-1$, proving their perpendicularity.

To write the equation of a line, first determine its slope ($m$). For a parallel line, use the same slope. For a perpendicular line, use the negative reciprocal slope. Then, use the point-slope formula with a known point $(x_1, y_1)$.

Find the line parallel to $y = 3x + 1$ through $(2, 4)$. The slope is $m=3$. Equation: $y - 4 = 3(x - 2)$, which simplifies to $y = 3x - 2$.
Find the line perpendicular to $y = -2x + 5$ through $(4, 0)$. The new slope is $m=\frac{1}{2}$. Equation: $y - 0 = \frac{1}{2}(x - 4)$, which simplifies to $y = \frac{1}{2}x - 2$.

Want to create a line that's a perfect partner to an existing one? First, find its slope—either the same for a parallel pal or the opposite reciprocal for a perpendicular rival. Then, grab the one point you know the new line passes through and plug it all into the point-slope formula to reveal the line's full equation!