Slope Criteria for Parallel and Perpendicular
Aligned to G-GPE.B.5 — Common Core State Standards for Mathematics.
What this lesson teaches
Two nonvertical lines are parallel exactly when their slopes are equal. They are perpendicular exactly when the product of their slopes is -1, meaning one slope is the negative reciprocal of the other. These criteria let you test and build lines with given relationships.
Worked examples
Example 1
Find a line perpendicular to y = 2x + 1 through the point (4, 3). The given slope is 2, so the perpendicular slope is the negative reciprocal, -1/2. Use point-slope form: y - 3 = -1/2 (x - 4). Distribute: y - 3 = -1/2 x + 2, so y = -1/2 x + 5. Its slope times the original is (-1/2)(2) = -1, confirming perpendicularity.
Example 2
Decide whether the quadrilateral A(0,0), B(4,2), C(6,6), D(2,4) is a parallelogram. Slope AB = 2/4 = 1/2 and slope DC = (6-4)/(6-2) = 2/4 = 1/2 — parallel. Slope BC = (6-2)/(6-4) = 4/2 = 2 and slope AD = 4/2 = 2 — parallel. Both pairs of opposite sides are parallel, so it is a parallelogram. It is not a rectangle, since (1/2)(2) = 1, not -1.
Example 3
Common mistake: calling the perpendicular slope the negative, or just the reciprocal. Perpendicular slopes multiply to -1, so you need the NEGATIVE RECIPROCAL: for slope 3/4 it is -4/3, not -3/4 and not 4/3. Two special cases break the rule: a horizontal line (slope 0) is perpendicular to a vertical line (slope undefined), and no reciprocal is involved — 0 has none.
Practice questions
- What is the slope of any line parallel to y = -3x + 7?
- Write the slope of a line perpendicular to a line with slope 4/5.
- Find the equation of the line through (0, 2) parallel to y = 5x - 3.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Criterios de pendiente para paralelas y perpendiculares
Dos líneas no verticales son paralelas exactamente cuando sus pendientes son iguales. Son perpendiculares exactamente cuando el producto de sus pendientes es -1, es decir, una pendiente es el recíproco negativo de la otra. Estos criterios permiten probar y construir líneas con relaciones dadas.
Ejemplo 1: Halla una línea perpendicular a y = 2x + 1 que pase por el punto (4, 3). La pendiente dada es 2, así que la pendiente perpendicular es el recíproco negativo, -1/2. Usa la forma punto-pendiente: y - 3 = -1/2 (x - 4). Distribuye: y - 3 = -1/2 x + 2, así que y = -1/2 x + 5. Su pendiente por la original es (-1/2)(2) = -1, confirmando la perpendicularidad.
Ejemplo 2: Decide si el cuadrilátero A(0,0), B(4,2), C(6,6), D(2,4) es un paralelogramo. Pendiente AB = 2/4 = 1/2 y pendiente DC = (6-4)/(6-2) = 2/4 = 1/2 — paralelas. Pendiente BC = (6-2)/(6-4) = 4/2 = 2 y pendiente AD = 4/2 = 2 — paralelas. Los dos pares de lados opuestos son paralelos, así que es un paralelogramo. No es rectángulo, porque (1/2)(2) = 1, no -1.
Ejemplo 3: Error común: llamar pendiente perpendicular al negativo, o solo al recíproco. Las pendientes perpendiculares multiplican -1, así que necesitas el RECÍPROCO NEGATIVO: para la pendiente 3/4 es -4/3, no -3/4 ni 4/3. Dos casos especiales rompen la regla: una recta horizontal (pendiente 0) es perpendicular a una vertical (pendiente indefinida), y ahí no hay recíproco alguno — 0 no lo tiene.
More Expressing Geometric Properties with Equations lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required