Slope, Distance and Midpoint Calculator

One pair of points, four answers

Give it two points on a coordinate plane, (x₁, y₁) and (x₂, y₂), and it returns everything about the segment between them in one pass: the slope, the equation of the line through them, the distance between them and the midpoint of the segment. Results are exact where the math allows, so a slope shows as 5/3 rather than 1.6666666667 and a distance as 3√13 before its decimal. It also works backward: from one endpoint and the midpoint, it finds the other endpoint.

How to use it

1. Choose Two points, or Endpoint and midpoint if you know the middle of the segment and one end.
2. Type each coordinate. Whole numbers, decimals, fractions such as 3/4, mixed numbers such as 2 1/2 and negatives all work, and they are read exactly: 0.1 is 1/10. A point copied as (3, 4) can go straight into an x box; the matching y box then disappears, and the line under the box says how the entry was read.
3. Decimal places sets the rounding of the decimal forms only. Exact fractions and square roots are never rounded.

The Try chips load a vertical line, a horizontal line, fraction coordinates, a 1 in 12 ramp and an endpoint-from-midpoint problem. Save for comparison keeps up to three results side by side, and for a sloped line the Triangle Calculator link carries the rise and run over as the legs of a right triangle.

How to find the slope from two points

Subtract the y-coordinates, subtract the x-coordinates in the same order, and divide:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

• $\Delta y = y_2 - y_1$ is the rise, the vertical change from point 1 to point 2.
• $\Delta x = x_2 - x_1$ is the run, the horizontal change.
• A positive slope rises from left to right, a negative one falls, and a bigger size means steeper.

Which point you call point 1 doesn’t matter, as long as both subtractions start from the same point: swapping the points flips the sign of the rise and the run together, and the quotient stays the same.

Worked example: (−4, 5) and (5, −1)

1. Rise: Δy = −1 − 5 = −6. Run: Δx = 5 − (−4) = 9.
2. Slope: m = −6 ÷ 9 = −2/3 ≈ −0.6667. Going right, the line falls 2 units for every 3 units across.
3. Angle of inclination: tan⁻¹(−2/3) = −33.6901°. Angles are measured counterclockwise from the positive x-axis, so θ = 180° − 33.6901° = 146.3099°. The percent grade is −2/3 × 100% = −66.67%.
4. Distance: d = √(9² + (−6)²) = √(81 + 36) = √117. Since 117 = 3² × 13, d = 3√13 ≈ 10.8167.
5. Midpoint: ((−4 + 5) ÷ 2, (5 + (−1)) ÷ 2) = (1/2, 2).
6. Line: the y-intercept is b = 5 − (−2/3)(−4) = 5 − 8/3 = 7/3, so y = −(2/3)x + 7/3. Moving the x-term gives (2/3)x + y = 7/3, and multiplying by 3 gives the standard form 2x + 3y = 7, which crosses the x-axis at x = 7/2.
7. Perpendicular bisector: slope 3/2, the negative reciprocal of −2/3, through (1/2, 2). In standard form that is 6x − 4y = −5.

Undefined vs zero slope: vertical and horizontal lines

A horizontal line has slope 0: the rise is 0, and 0 divided by any run is 0. Through (−5, 4) and (7, 4) the line is y = 4, with an angle of 0° and a grade of 0%.

A vertical line’s slope is not infinite; it doesn’t exist. Undefined is the word textbooks use: the run is 0, and division by 0 has no answer. Through (3, −2) and (3, 6) the line is x = 3. It has no slope-intercept or point-slope form, it never meets the y-axis, and its perpendicular bisector is the horizontal line y = 2 through the midpoint (3, 2). The distance is simply the difference in y, 8.

Horizontal and vertical lines are perpendicular even though the “product of slopes is −1” rule can’t be applied to them, because one of the slopes doesn’t exist.

The same point typed twice is a third special case. Every line through that point passes through “both” points, so there is no single slope or equation; the distance is 0 and the midpoint is the point itself.

Equation of a line: slope-intercept, point-slope and standard form

The same line can be written three common ways. For the worked example:

To turn slope-intercept form into standard form, move the x-term to the left, multiply both sides by the common denominator to clear the fractions, and multiply by −1 if the x-coefficient is negative. This page writes standard form with whole numbers that share no common factor and a positive A (a positive B for a horizontal line), the usual preference for standard form. With the Fractions chip, (1/2, 3/4) and (2 1/2, −1/4), the steps give y = −(1/2)x + 1 and then x + 2y = 2.

Distance formula between two points

The distance is the length of the hypotenuse of the right triangle whose legs are the run and the rise:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

To simplify the root, pull out the largest perfect-square factor: √117 = √(9 × 13) = 3√13. A number with no square factor stays as it is, so the distance from (0, 0) to (2, 5) is √29 ≈ 5.3852. Squaring removes the signs, so it doesn’t matter which point comes first.

Midpoint formula (and finding a missing endpoint)

The midpoint of a segment averages the coordinates of its endpoints:

$M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$

Solving that for point 2 finds a missing endpoint: $x_2 = 2x_M - x_1$ and $y_2 = 2y_M - y_1$. If one endpoint is (2, −3) and the midpoint is (5, 1), the other endpoint is (2 × 5 − 2, 2 × 1 − (−3)) = (8, 5). Check it by averaging: ((2 + 8) ÷ 2, (−3 + 5) ÷ 2) = (5, 1). That segment has slope 4/3 and length 10.

Slope as an angle, percent grade or ratio

Outside a math class, the same steepness is often given in one of three other ways:

• Angle of inclination: θ = tan⁻¹(m), measured counterclockwise from the positive x-axis, from 0° up to 180°. A falling line gets an angle over 90°.
• Percent grade: 100 × m. A rise of 1 over a run of 12 is 8.33%.
• Ratio: rise : run, often written “1 in N” with N = run ÷ rise. The U.S. Access Board’s guide to the ADA Standards writes ramp slopes this way, as 1:12, and gives the same slope as 8.33%.

So a 10% slope is about 5.71°, and a 1 to 5 slope is about 11.31°. A 100% grade is 45°, not a vertical wall: a vertical line has no percent grade at all.

Reading the result

• Headline: the slope as an exact fraction with its decimal, or the other endpoint when you started from a midpoint.
• Tiles: distance, midpoint, rise over run, angle of inclination, percent grade with the rise : run ratio, and the perpendicular bisector of the segment.
• Equation of the line: slope-intercept, point-slope, standard and general forms, both intercepts and the slope of perpendicular lines.
• Graph: both points, the segment and the line, the run and rise as dashed legs and the midpoint as a hollow dot, drawn with the same scale on both axes so the angle looks right. The key-points table under it lists the same points and downloads as CSV.
• Steps: one list per result with your numbers substituted, plus the spreadsheet formulas =SLOPE() and =SQRT() that give the same decimals.

Assumptions and limitations

• Both axes are in the same unit. The slope is correct in any units, but the angle and percent grade only mean something physical when x and y are measured the same way.
• Distance is on a flat plane. Latitude and longitude are angles on a sphere, so the distance formula doesn’t turn them into miles or kilometers.
• Coordinates can be between −1,000,000,000,000 and 1,000,000,000,000, and non-zero ones at least 0.000000000001 in size. All arithmetic is exact; a square root is shown in simplified form when its numbers are small enough to factor quickly, and as a decimal otherwise.

Common mistakes

• Mixing the order of subtraction: (y₂ − y₁) ÷ (x₁ − x₂) flips the sign of the slope. Start both subtractions from the same point.
• Run over rise: dividing Δx by Δy gives the reciprocal, −3/2 instead of −2/3 in the worked example.
• Calling a vertical slope infinity or 0: no number fits. Undefined is the answer, the line is written x = a, and zero belongs to horizontal lines.
• Losing a sign when squaring: (−6)² is 36, so the example’s distance is √(81 + 36), not √(81 − 36).
• Subtracting in the midpoint formula: the midpoint adds the coordinates before halving; (x₂ − x₁) ÷ 2 is half the run, not a coordinate.
• Leaving fractions or a negative A in standard form: (2/3)x + y = 7/3 and −2x − 3y = −7 describe the same line as 2x + 3y = 7, but only the last one is in the usual standard form.