Calculadora de inclinação de retas paralelas

This calculator finds the equation of a line that is parallel to a given line and passes through a given point. The key fact: parallel lines have equal slopes. So if you know the slope of any line and a point on the new line, you can write its equation directly via the point-slope formula. This tutorial covers the parallel and perpendicular slope rules, three worked examples, and the geometric reasoning behind them.

The parallel slope rule

Two non-vertical lines are parallel if and only if they have the SAME slope:

m₁ = m₂

(Two vertical lines are also parallel — they share the "undefined slope" classification.)

The geometric reason: slope measures how steeply a line rises per unit of horizontal distance. Two lines with the same slope rise at the same rate, so they maintain a constant vertical distance — never meeting.

The perpendicular slope rule

Two non-vertical lines are perpendicular if and only if their slopes multiply to −1:

m₁ × m₂ = −1

Equivalently, each slope is the "negative reciprocal" of the other: m₂ = −1/m₁.

Special case: a horizontal line (slope 0) is perpendicular to a vertical line (undefined slope). The "negative reciprocal of 0" doesn't make algebraic sense, but the geometric perpendicular relation still holds.

Worked example 1 — line parallel to a given line

Given line: y = 2x + 3. Find the equation of the line parallel to this through point (4, 5).

Step 1: Identify the slope. From y = mx + b form: m = 2.

Step 2: Use the parallel rule. The new line has the same slope, m = 2.

Step 3: Apply point-slope form: y − 5 = 2(x − 4).

Step 4: Simplify to slope-intercept: y = 2x − 8 + 5 = y = 2x − 3.

Worked example 2 — from given slope value

Find the line with slope 3/4 passing through point (−2, 1), parallel to a previously-defined line of the same slope.

By the parallel rule, any line with slope 3/4 is parallel to any other line with slope 3/4. The equation: y − 1 = (3/4)(x − (−2)) = (3/4)(x + 2).

Slope-intercept: y = (3/4)x + 3/2 + 1 = y = (3/4)x + 5/2.

Worked example 3 — perpendicular line

Find the line perpendicular to y = (2/3)x − 1, passing through point (3, 4).

Step 1: Slope of given line: m₁ = 2/3.

Step 2: Perpendicular slope: m₂ = −1 / (2/3) = −3/2.

Step 3: Point-slope: y − 4 = (−3/2)(x − 3).

Step 4: Slope-intercept: y = (−3/2)x + 9/2 + 4 = y = (−3/2)x + 17/2.

Why parallel slopes are equal

The slope of a line in the form y = mx + b is the ratio of "rise over run" — the change in y per unit change in x. Two lines with the same slope have the same "lean".

If two lines have different slopes (say m₁ < m₂), they grow at different rates. At some x-value, the gap between them closes to 0 — they intersect. Lines with the same slope maintain a constant gap and never intersect (unless they're the same line).

Why perpendicular slopes multiply to −1

Suppose line ℓ has slope m. Rotate ℓ by 90° (counterclockwise) — the rotated line is perpendicular to ℓ.

Under a 90° rotation, the point (1, m) (one step right, m steps up from origin along ℓ) maps to (−m, 1) (the rotation formula). The new line passes through the origin and (−m, 1), giving slope 1/(−m) = −1/m.

So the perpendicular line has slope −1/m. Multiplying: m × (−1/m) = −1.

The two forms of the line equation

Point-slope form: y − y₁ = m(x − x₁). Use this when you know a point (x₁, y₁) and the slope m. Direct write-up — no algebra needed.

Slope-intercept form: y = mx + b. Use this when you know the slope m and the y-intercept b. Easier to graph and evaluate.

Both forms are equivalent — they describe the same line. Convert from point-slope to slope-intercept by distributing m and combining the constants.

Real-world applications

• Architectural drafting. Walls, beams, and rafters often need to be parallel — drawing them computationally requires the parallel slope rule.
• Road engineering. Highway lanes, runways, train tracks are all designed with parallel constraints.
• Computer graphics. Aligning UI elements (text, buttons, columns) uses parallel-line geometry.
• Physics — kinematics. Objects with parallel velocity vectors never collide; perpendicular vectors maximally diverge.
• Crystallography. Crystal lattice planes are families of parallel planes — slope relationships are foundational.

Common mistakes

• Using a different slope for the parallel line. Parallel = SAME slope. Different = not parallel.
• Confusing parallel and perpendicular. Parallel = equal slopes (m₁ = m₂). Perpendicular = negative reciprocal slopes (m₁ × m₂ = −1).
• Forgetting the negative sign for perpendicular. The reciprocal is 1/m, but the perpendicular slope is −1/m. Forgetting the minus gives a different line.
• Handling vertical lines. A vertical line (x = c) has undefined slope. Its parallel is also vertical (x = different c). Its perpendicular is horizontal (y = c) with slope 0. The standard rules don't apply directly because of the undefined slope.