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Two lines in a plane are parallel exactly when they share the same slope — no matter how far apart they sit, they never converge or diverge, always staying the same perpendicular distance apart. Given an existing line and a point that's not already on it, there is exactly one line through that point which is parallel to the original — this calculator finds its equation directly. Finding a parallel line through a given point is a routine task in drafting and CAD software (extending a wall, road, or structural member so it stays aligned with an existing one), in physics (constructing a line representing a second object's motion that shares a first object's velocity direction), and in geometry proofs and constructions that rely on the parallel postulate, such as proving properties of parallelograms or applying the alternate-interior-angles theorem.
How it works
Enter two points that define the original line, plus the new point the parallel line must pass through. The calculator first finds the original line's slope using the two given points: m = (y2 − y1)/(x2 − x1). Because parallel lines always share the same slope, that same m is then plugged into point-slope form, y − y1 = m(x − x1), using the new point's coordinates in place of (x1, y1) — giving the equation of the one and only line through the new point that runs parallel to the original.
- Enter original line — point 1 x1.
- Enter original line — point 1 y1.
- Enter original line — point 2 x2.
- Enter original line — point 2 y2.
- Enter new point — x.
- Enter new point — y.
- Click Calculate to see your results.
Examples
Line through (0,0), (2,4); new point (3,1)
Slope = (4−0)/(2−0) = 2. Using point-slope form through (3,1): y − 1 = 2(x − 3), which simplifies to y = 2x − 5.
Line through (1,5), (4,-1); new point (0,0)
Slope = (-1−5)/(4−1) = -6/3 = -2. Using point-slope form through (0,0): y − 0 = -2(x − 0), which simplifies to y = -2x.
Vertical line through (3,0), (3,7); new point (-2,5)
The original line is vertical (undefined slope, since x doesn't change). Its parallel counterpart is also vertical, taking the form x = -2, since every vertical line has the equation x = (a constant).
Common mistakes to avoid
- Using the new point's coordinates instead of the original line's two points when computing the slope.
- Forgetting the special vertical-line case, where slope is undefined and the answer must be written as x = constant instead of using point-slope form.
- Mixing up which point is the "new point" (the one the answer must pass through) and which two points merely defined the original line's slope.
- Confusing this calculation with finding a perpendicular line — parallel keeps the same slope, perpendicular flips and inverts it.