7.2 Equations of Parallel & Perpendicular Lines

Slope Criteria for Parallel and Perpendicular Lines

Standard GEO-G.GPE.5 asks you to prove and apply the slope criteria for parallel and perpendicular lines, then write equations that satisfy them. Two facts drive every question:

• Parallel lines have equal slopes (and different y-intercepts, or they would be the same line): m₁ = m₂.
• Perpendicular lines have slopes that are negative reciprocals, so their product is −1: m₁ · m₂ = −1.

The one exception is the horizontal-vertical pair: a horizontal line (slope 0) is perpendicular to a vertical line (undefined slope), even though you cannot multiply an undefined value.

Three Forms of a Line

Being fluent in all three matters because the Regents mixes them across the answer choices. From standard form, extract the slope quickly as −A/B. For example, 3x − 4y = 8 rearranges to y = (3/4)x − 2, so its slope is 3/4 — or just compute −A/B = −3/(−4) = 3/4.

Finding the Right Slope

Negative reciprocal means two steps, and skipping either is the classic error: flip the fraction and change the sign. The negative reciprocal of 3/4 is −4/3; of −2 (that is, −2/1) it is 1/2; of 5 it is −1/5.

Writing Equations Through a Given Point

Most Regents items give you a line and a point, then ask for a parallel or perpendicular line through that point. Point-slope form is almost always the fastest tool.

Worked example — perpendicular: Line l has equation 3x − 4y = 8. Write a line perpendicular to l through (2, −1). First find the slope of l: rearranging gives y = (3/4)x − 2, so m = 3/4. The perpendicular slope is the negative reciprocal, −4/3. Substitute the point into point-slope form: y − (−1) = −4/3(x − 2), which is written y + 1 = −4/3(x − 2).

Worked example — parallel: Write a line parallel to y = −2x + 5 through (1, 4). Parallel means the same slope, −2. Point-slope gives y − 4 = −2(x − 1), which simplifies to y = −2x + 6. Note the y-intercept changed from 5 to 6 — a parallel line must have a different intercept, confirming it is not the original line.

Parallel, Perpendicular, or Neither

A common multiple-choice task gives two lines and asks how they relate. Convert both to slope-intercept form, compare slopes, and decide: equal slopes mean parallel, slopes that multiply to −1 mean perpendicular, and anything else means the lines intersect but not at a right angle.

Worked example: How do 2x − 3y = 12 and 3x + 2y = 4 relate? The first gives y = (2/3)x − 4 (slope 2/3); the second gives y = −(3/2)x + 2 (slope −3/2). Their product is (2/3)(−3/2) = −1, so the lines are perpendicular.

Horizontal and Vertical Special Cases

Horizontal and vertical lines break the reciprocal shortcut, so handle them directly. A line parallel to y = 4 through (2, 7) is y = 7. A line perpendicular to y = 4 through (2, 7) is the vertical line x = 2. Likewise, a line perpendicular to x = −3 through (5, 1) is the horizontal line y = 1. When one slope is 0 and the other is undefined, the lines are still perpendicular even though the product rule cannot be applied.

Why the Criterion Works

The perpendicular rule is not arbitrary. Rotating a line 90° about the origin sends a direction of "run a, rise b" to "run −b, rise a," so a slope of b/a becomes a/(−b) = −a/b — precisely the negative reciprocal. GEO-G.GPE.5 expects you to justify this relationship, not only apply it.

Matching the Answer Form

The correct line may appear in point-slope, slope-intercept, or standard form among the choices. From y + 1 = −4/3(x − 2), distribute to y = −4/3 x + 5/3, then clear fractions to 4x + 3y = 5. All three describe the same line, so if your point-slope answer is missing, convert before eliminating choices.

Testing Whether Lines Meet at a Right Angle

Coordinate proofs (Section 7.4) lean on the perpendicular test to prove right angles. Given two segments, compute both slopes and multiply. If the product is exactly −1, the segments are perpendicular and form a right angle; if the slopes are equal, the segments are parallel; otherwise they are neither.

Common traps

• Forgetting the sign when taking a reciprocal: the perpendicular of 3/4 is −4/3, not 4/3.
• Only flipping or only negating instead of doing both.
• Reading the slope straight off standard form as A/B instead of −A/B.
• Claiming two lines with slope 2 are perpendicular — equal slopes are parallel, not perpendicular.
• Writing a "parallel" line that is actually identical because you kept the same y-intercept.