Use slope criteria to determine whether lines are parallel, perpendicular, or neither, and write equations of such lines (NC.M1.G-GPE.5).

What this topic is asking

NC.M1.G-GPE.5 asks you to use slope criteria to decide whether two lines are parallel, perpendicular, or neither, and to solve geometric problems involving slopes and equations of lines, including writing the equation of a line parallel or perpendicular to a given one through a given point.

The two slope criteria

These two facts are the whole standard.

To get a negative reciprocal: flip the fraction and switch the sign. The negative reciprocal of $4$ (that is $\frac{4}{1}$) is $-\frac{1}{4}$.

Writing a parallel line

Writing a perpendicular line

For a perpendicular line, swap to the negative reciprocal slope first.

To write the line through $(4, 1)$ perpendicular to $y = \frac{1}{2}x + 3$: the given slope is $\frac{1}{2}$, so the perpendicular slope is $-2$ (flip to $\frac{2}{1}$, negate). Then point-slope: $y - 1 = -2(x - 4)$, giving $y = -2x + 9$.

Deciding from two equations

Given two lines, compare their slopes:

• Same slope, different intercept: parallel.
• Slopes multiply to $-1$: perpendicular.
• Neither: the lines intersect at some non-right angle.

For $y = 3x + 1$ and $y = -\frac{1}{3}x + 5$: slopes $3$ and $-\frac{1}{3}$ multiply to $-1$, so they are perpendicular.

How the NC Math 1 EOC examines this topic

• Multiple choice. Identify whether lines are parallel, perpendicular, or neither, or choose a perpendicular slope.
• Gridded response. Write or evaluate a parallel or perpendicular line.
• Technology-enhanced. Match equations to a described relationship.

This applies the slope and line-writing skills to geometry and underlies coordinate proofs, where slopes prove sides are parallel or perpendicular.

Why slope encodes direction

Slope is more than steepness, it is the direction of a line. Two lines point the same way exactly when their slopes match, which is why equal slopes mean parallel. Perpendicularity is a quarter-turn, and turning a direction $90$ degrees flips rise and run and reverses one sign, which is precisely the negative-reciprocal rule. The product being $-1$ is the algebraic fingerprint of that right angle. Seeing slope as direction explains both criteria at once and makes them memorable: parallel keeps the direction, perpendicular rotates it a quarter turn. This single idea powers coordinate proofs about rectangles, right triangles, and parallelograms, where the whole argument rests on comparing slopes.

Try this

Q1. Are $y = 4x - 1$ and $y = 4x + 6$ parallel, perpendicular, or neither? [1 point]

• Cue. Equal slopes ($4$): parallel.

Q2. Write the line through $(0, -2)$ perpendicular to $y = \frac{1}{3}x + 4$. [2 points]

• Cue. Perpendicular slope is $-3$; through $(0, -2)$: $y = -3x - 2$.