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CalcMax

Endpoint Calculator

Range: -1,000,000,000 – 1,000,000,000

Range: -1,000,000,000 – 1,000,000,000

Range: -1,000,000,000 – 1,000,000,000

Range: -1,000,000,000 – 1,000,000,000

Result

5.0000

Other endpoint x

Other endpoint y
8.0000
Distance from the midpoint to either end
3.6056

An endpoint calculator runs the midpoint calculation backwards. You know the middle of a segment and one of its ends, and you want the other end. The rule is a subtraction: double the midpoint coordinate and take the known endpoint away, once for each axis. If the midpoint is (3, 5) and one end is (1, 2), the other end is (5, 8), because twice three minus one is five and twice five minus two is eight. The answer is the missing end, and the two coordinates beside it are the confirmation that the reversal was done correctly. That confirmation is this page's third number, and it is the one worth understanding. The midpoint is by definition the same distance from both ends, so the gap from the known endpoint to the midpoint has to equal the gap from the midpoint to the endpoint you just recovered. The page prints the first of those two gaps. If it matches the half of the segment you already knew by hand, the answer is right; if the recovered end is on the wrong side of the midpoint, the printed half-length will still look perfectly reasonable, and the way to catch it is to check that the point sits on the opposite side of the midpoint from the end you entered. Two things about the inputs are worth knowing first. The coordinates are plain numbers with no unit attached, so nothing on the page is in centimetres or metres: if your numbers came off a map in metres, the answer is in metres, and if they came off a graph, they are in whatever the grid was. And the four boxes accept anything between minus a billion and a billion, but the recovered endpoint is not held to that range — twice the midpoint minus the known end can reach three billion on the far side, and the page will print it rather than complain. That is not an overflow; it is an ordinary subtraction producing a number larger than either of the numbers it was built from.

Missing endpoints for segments you are likely to meet

Midpoint xMidpoint yKnown xKnown yOther xOther yHalf-length
3512583.6056
0034-3-45
-214-3-857.2111
2222220
0000000
1000000000200000001000000
555-551510
-1.52.53.5-0.5-6.55.55.831

Eight segments, and the last three columns are the page's answer plus its own check. The first row is the input the page loads with, and it is the same segment the midpoint page loads with read the other way round: that page takes (1, 2) and (5, 8) and returns (3, 5), and this one takes (3, 5) and (1, 2) and returns (5, 8). It is the cheapest check that both formulas are wired the right way round, and it works on any pair rather than only on this one. The second row has the midpoint at the origin, which makes the rule collapse to a change of sign, and the third has the known end in one quadrant and the answer in the diagonally opposite one, where subtracting a negative coordinate is what moves the answer past the midpoint instead of toward it. The fourth row is a segment of no length, the one case where the answer is not opposite the known end. The fifth row has every coordinate zero, which is a real answer rather than an empty one. The sixth is the row that shows an answer outside the range of the inputs — a midpoint of one million from the origin with a known end at the origin gives an answer two million out — and the last two rows are a vertical segment and a fractional one, where the half-length of 5.831 is a square root and therefore rounded.

Formula

Other endpoint x = 2 × (midpoint x) − (known endpoint x) Other endpoint y = 2 × (midpoint y) − (known endpoint y) Half-length = √((known x − midpoint x)² + (known y − midpoint y)²)

Midpoint
An x and a y for the point in the middle of the segment. It is given to you rather than computed, so it must be the true midpoint of the segment you have in mind — the page has no way to tell that it is not
Known endpoint
The end you already have. It must be one of the two ends and not some other point on the segment, since the rule assumes it is exactly as far from the midpoint as the answer will be
Other endpoint
The answer, one coordinate per axis, each rounded to four decimal places. It lies the same distance from the midpoint as the known end does, but on the opposite side, so it is usually outside the range of the numbers you typed in
Half-length
The distance from the known endpoint to the midpoint, worked out from those two points directly. It is the page's check on itself: the recovered end has to sit the same distance from the midpoint, and comparing this number against your own measurement is what catches a reversed sign
Four decimal places
The width every number is written to, trailing zeros included. Coordinates that are whole numbers still print with four zeros after them, which is a display choice rather than a claim about precision

The obvious job is finishing a shape: you have the centre of something and one edge, and you need the point directly opposite. The centre of a circle and one end of a diameter gives you the other end of that diameter, which is the same problem in different words. A surveyor who has set a stake at the middle of a run and knows where one end landed wants the other end, and that is this page exactly. It comes up in drawing too — the middle of a line and one end, and you want the far end to place a mirror or a matching feature. There is a second use that is less obvious and often more valuable: checking. Given a segment, take its midpoint, feed that and one end in here, and the page should hand back the other end. If it does not, the midpoint was wrong or the segment is not straight, and either way you have found something worth knowing before you build on it. The half-length row is the quickest version of that check, since a shape whose two halves disagree is not symmetric about the point you thought was its middle. One habit makes the answer much easier to trust: before reading the result, ask which side of the midpoint the new point should be on. The answer is always the opposite side from the endpoint you entered, and a result on the same side as the known end is a sign error rather than a surprise.

Worked examples

  1. The default pair, which is the midpoint page's default read backwards

    1. Double the midpoint x: 3 × 2 = 6, then take the known x away: 6 − 1 = 5
    2. Double the midpoint y: 5 × 2 = 10, then take the known y away: 10 − 2 = 8
    3. The other endpoint is (5, 8)
    4. Half-length: the gap from (1, 2) to (3, 5) is √(4 + 9) = √13 = 3.6056

    The same segment the midpoint page loads with, run in the other direction: feed that page (1, 2) and (5, 8) and it returns (3, 5), which is where this page starts. The two pages agreeing is the cheapest possible check that neither formula has its signs reversed, and it works on any input rather than only on this one. The half-length of 3.6056 is the same number the midpoint page prints for the same segment, which is not a coincidence — it is the same gap measured from the same pair of points.

  2. The other end of a segment whose middle is the origin

    1. Double the midpoint x: 0 × 2 = 0, then take the known x away: 0 − 3 = −3
    2. Double the midpoint y: 0 × 2 = 0, then take the known y away: 0 − 4 = −4
    3. The other endpoint is (−3, −4)
    4. Half-length: the gap from (3, 4) to the origin is √(9 + 16) = 5

    A midpoint at the origin makes the rule collapse to a change of sign, and that is the clearest case for seeing what the page does: the answer is a reflection of the known end through the midpoint. With several zeros in the input it is also easy to lose a sign, so this is the input to reach for if a result looks wrong — the answer here is forced and there is nothing to round.

  3. An endpoint in the opposite quadrant

    1. Double the midpoint x: (−2) × 2 = −4, then take the known x away: −4 − 4 = −8
    2. Double the midpoint y: 1 × 2 = 2, then take the known y away: 2 − (−3) = 5
    3. The other endpoint is (−8, 5)
    4. Half-length: the gap from (4, −3) to (−2, 1) is √(36 + 16) = √52 = 7.2111

    Every sign is doing something here, and the second line is the one that catches people: subtracting a negative y adds, so the answer of 5 is above the midpoint of 1 and not below it. The recovered end sits in the diagonally opposite quadrant from the known end, which is the general pattern — a segment through a midpoint crosses from one side to the other in both axes at once.

  4. A known endpoint that is the midpoint itself

    1. Double the midpoint x: 2 × 2 = 4, then take the known x away: 4 − 2 = 2
    2. The same for y: 4 − 2 = 2
    3. The other endpoint is (2, 2), the point you started from
    4. Half-length: the gap is zero, because the two points coincide

    A segment of zero length is the one case where the answer is not on the far side of the midpoint — it is the midpoint, and the known end is there too. That is the correct answer rather than a degenerate failure: if both ends of a segment are the same point, then that point is also its middle, and the page has nothing else to say. Worth seeing once so that a repeated coordinate does not read as a bug.

  5. A midpoint a long way from the known end

    1. Double the midpoint x: 1000000 × 2 = 2000000, then take the known x away: 2000000 − 0 = 2000000
    2. Double the midpoint y: 0 × 2 = 0, then take the known y away: 0 − 0 = 0
    3. The other endpoint is (2000000, 0)
    4. Half-length: the gap from the origin to (1000000, 0) is 1000000

    The case that shows the answer is not confined to the range of the inputs. Both numbers typed in are within a million of the origin, and the answer is two million out — further than anything on the page, because the known end and the answer straddle the midpoint rather than sitting near each other. Nothing here is an overflow or an approximation: doubling and subtracting whole numbers stays exact well past this size.

Limitations

The page assumes the point you call the midpoint really is one. It does not verify that the known end and the recovered end are equally far from it — it constructs the second one so that they are, which means a midpoint that was wrong going in produces an answer that is wrong by exactly the same amount, and the page cannot tell. If the midpoint you have is an estimate rather than a measurement, so is the answer. The coordinates carry no unit, and the page neither converts nor stores one, so it cannot see that a midpoint in metres and an endpoint in millimetres were mixed. The four inputs are limited to plus or minus one billion, but the recovered endpoint is not: it can reach three billion, and that is correct rather than a bug, because doubling then subtracting is allowed to overshoot the numbers it was given. A recovered coordinate outside the range you typed is normal on this page. The half-length is rounded to four decimal places and is often irrational, so it is an approximation for most inputs, and a hand check against your own measurement will not always agree to the last digit. Finally, the page recovers a point and says nothing else: it does not give you the full length of the segment, the slope of the line through it or a third dimension, so a pair of points in space cannot be completed here even if two of the three coordinates are known.

Frequently asked questions

What is the endpoint formula?
Double the midpoint and subtract the endpoint you know, once for each axis. If the midpoint is 3 and the known end is 1, the other end is 6 minus 1, which is 5. It is the midpoint rule rearranged: the midpoint is the average of the two ends, so the two ends added together are twice the midpoint, and taking one of them away leaves the other. There is nothing extra to remember beyond the direction of the last subtraction.
Why is the answer outside the range of the numbers I typed in?
Because the two ends sit on opposite sides of the midpoint, so the answer is as far past the middle as the end you entered is behind it. Type in a midpoint a million from the origin and a known end at the origin, and the answer is two million out. That is not an overflow or a mistake — doubling and subtracting is allowed to produce a larger number than either input, and the page has no upper limit of its own for the result, only the billion-wide limit on what you can type into the four boxes.
What is the third number on the page for?
It is the page checking itself. A midpoint is the same distance from both ends, so the gap between the known endpoint and the midpoint has to equal the gap between the midpoint and the endpoint you just recovered. The page prints the first of those two gaps, and if it matches the half of the segment you already knew, the answer is consistent. Comparing against the second gap would prove nothing, since the page would be using its own answer to test its own answer.
Does the page check that my midpoint is really the midpoint?
No, and it cannot. The page uses the point you call the midpoint as the starting fact and builds the other end so that it sits the same distance away — so if the point you entered was not the true middle of the segment, the answer will be wrong by exactly the same amount and nothing on the page will look unusual. A midpoint that is an estimate rather than a measurement gives an estimated endpoint. The half-length row is the only handle you get, and it only tells you about the geometry you entered, not about the world.
Can I use this for points in space?
Only if you work one axis at a time. The page has two coordinate boxes per point and no third, so a three-dimensional problem cannot be entered as it stands. The rule itself has nothing to do with two dimensions — double the midpoint and subtract the known end, on every axis — so you can apply the same arithmetic to the third coordinate yourself, but the page will not do it and will not show the result.
What if the known endpoint is the midpoint?
Then the answer is that same point, and the half-length is zero. A segment whose ends coincide has no length, and its middle is the point itself, so nothing has gone wrong. It is the one case where the recovered end is not on the opposite side of the midpoint, and seeing it once is worth the trouble so that a repeated coordinate does not look like a failure of the calculation.

References

Related calculators

Midpoint CalculatorMidpoint of a line segment between two coordinates.Distance CalculatorDistance between two points, with the working shown.Pythagorean Theorem CalculatorSolve any side of a right triangle from the other two.Rise Over Run CalculatorSlope, angle and percent grade from two points.