Skill: parallel-perpendicular

Parallel & Perpendicular Lines

Slopes control relationships: parallel lines share a slope; perpendicular slopes are opposite reciprocals. Learn to spot both on sight and to write the equation of the parallel or perpendicular through any point.

1 Slope relationships

Two lines are parallel when their slopes are equal; perpendicular when their slopes are opposite reciprocals (flip the fraction and change the sign).

The two rules

parallel: m₁ = m₂ · perpendicular: m₁ · m₂ = −1

2/3 is perpendicular to −3/2. A slope of 2 is perpendicular to −1/2. Horizontal (m = 0) and vertical (undefined) lines are perpendicular to each other — the one special case.

Left: equal slopes never meet. Right: slopes 2 and −1/2 meet at a right angle (2 · −1/2 = −1).

Writing the equation: keep the (new) slope, then plug the point into y = mx + b to find b. Parallel keeps m; perpendicular flips it to −1/m.

2 Worked examples

Identify, then write.

Example 1 Spot the parallels

Are y = 2x + 1 and y = 2x − 5 parallel?

Both slopes are 2 — yes, parallel. (Different intercepts, so not the same line.)

Example 2 Spot the perpendiculars

Are lines with slopes 2/3 and −3/2 perpendicular?

Flip 2/3 to 3/2 and change the sign: −3/2. Yes — opposite reciprocals.

Example 3 Perpendicular slope

Find the slope perpendicular to m = 4.

−1/4 = −1/4 — flip (4 = 4/1) and change the sign.

Example 4 Perpendicular through a point

Write the equation perpendicular to y = 2x + 1 through (4, 2).

New slope: −1/2. Plug in: 2 = −(1/2)(4) + b gives b = 4.

So y = −(1/2)x + 4.

3 Common mistakes

Three traps, each with the wrong version and the fix.

1. Using the same slope for perpendicular lines

Wrong: the perpendicular to y = 2x + 1 also has slope 2. Right: perpendicular slopes are opposite reciprocals: −1/2.

2. Flipping without changing the sign

Wrong: perpendicular to 2/3 is 3/2. Right: flip and change sign: −3/2. Both moves, every time.

3. Forgetting horizontal/vertical

Wrong: “the perpendicular to y = 3 has slope −1/0.” Right: horizontal (m = 0) meets vertical (undefined slope): the perpendicular to y = 3 is x = a.

4 Quick checks

Try these yourself, then reveal the answer.

Are y = −x + 2 and y = −x − 7 parallel, perpendicular, or neither?

Both slopes are −1: parallel.

Find the slope perpendicular to m = −2/5.

Flip to −5/2, change sign: 5/2.

Write the parallel to y = 3x − 2 through (0, 5).

Keep m = 3; the point (0, 5) gives b = 5: y = 3x + 5.

5 Key points

Remember

Parallel: equal slopes (m₁ = m₂).
Perpendicular: opposite reciprocals (m₁ · m₂ = −1).
Flip and change the sign — both moves, every time.
To write the equation: take the new slope, plug the point into y = mx + b, solve for b.
Horizontal ⊥ vertical is the special case: m = 0 meets undefined slope.

Key vocabulary

Parallel lines

Lines with equal slopes that never intersect.

Perpendicular lines

Lines meeting at a right angle; slopes multiply to −1.

Opposite reciprocal

Flip the fraction and change the sign: 2/3 ↔ −3/2.

Coincident lines

Same slope and same intercept — the “same line twice”, not just parallel.

Right angle

A 90° angle; the mark of perpendicular lines.

Practice relationships Open the line tool Explore in the lab