How to find endpoint given midpoint

Many students assume that if they know the midpoint and one endpoint, they can just average the coordinates they have to find the other endpoint. That is wrong. Averaging the known midpoint and endpoint will not give the missing endpoint; it will give a point that is one-quarter of the way along the segment, not the endpoint. To find endpoint given midpoint, you must reverse the midpoint formula. The midpoint is the average of the two endpoints. To undo an average, you multiply by 2 and subtract the known endpoint. This is the missing endpoint formula: B = (2M_x - A_x, 2M_y - A_y). This shows you how to solve the reverse midpoint problem, step by step, with algebra and a counting method on a graph, and covers the three common mistakes that cause wrong answers.

The missing endpoint formula is the algebraic rearrangement of the midpoint formula. Start with the midpoint formula for 2D: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). If you know M and one endpoint, call it A = (x₁, y₁), then solve for the other endpoint B = (x₂, y₂).

For the x-coordinate: M_x = (x₁ + x₂)/2. Multiply both sides by 2: 2M_x = x₁ + x₂. Subtract x₁: x₂ = 2M_x - x₁. The same logic applies to the y-coordinate: y₂ = 2M_y - y₁. For 3D problems, add the z-coordinate: z₂ = 2M_z - z₁.

This works for any two points in a Cartesian coordinate system, including those with negative and fractional coordinates. The formula is part of the Common Core standard HSG-GPE.B.6, which defines partitioning a directed line segment. The ratio division formula (which extends this to ratios other than 1:1) and the distance formula are related concepts, but the missing endpoint formula is the specific case for a 1:1 ratio.

To find the missing endpoint using algebra, set up two separate equations, one for each coordinate. This method is exactly the same as using the missing endpoint formula, but writing it out explicitly helps avoid arithmetic errors.

Step 1: Write the midpoint formula for the x-coordinate.
M_x = (x₁ + x₂)/2. Substitute the known values for M_x and x₁.

Step 2: Solve for x₂.
Multiply both sides by 2: 2M_x = x₁ + x₂. Subtract x₁: x₂ = 2M_x - x₁.

Step 3: Repeat for the y-coordinate (and z-coordinate in 3D).
M_y = (y₁ + y₂)/2. Solve: y₂ = 2M_y - y₁.

Step 4: Write the missing endpoint as an ordered pair.
B = (x₂, y₂). Check your work by plugging both endpoints into the midpoint formula: the average should equal the given midpoint.

This method is the same as the missing endpoint formula but forces you to show each operation. It is the safest approach for students who rush. The failure case: forgetting to multiply the midpoint by 2 before subtracting. If you only subtract the known endpoint from the midpoint, you will get a point that is not an endpoint.

If you have a graph and do not want to use algebra, you can find the missing endpoint by counting. This method works for integer coordinates and is useful for visual learners who are finding the midpoint of a segment on a coordinate grid.

Step 1: Plot the known endpoint and the midpoint. Mark them on the graph.

Step 2: Count the horizontal and vertical distances from the known endpoint to the midpoint. For example, if the known endpoint is at (1, 2) and the midpoint is at (4, 5), the horizontal distance is +3 (from x=1 to x=4) and the vertical distance is +3 (from y=2 to y=5).

Step 3: Apply those same distances from the midpoint to locate the missing endpoint. Since the midpoint is halfway, the distances from the midpoint to the missing endpoint are the same as from the known endpoint to the midpoint. From (4, 5), add +3 horizontally to get x=7, and add +3 vertically to get y=8. The missing endpoint is (7, 8).

This counting method is a direct application of the definition of a midpoint: the point that splits a line segment into two equal lengths. It works because the changes in x and y from one endpoint to the midpoint are exactly half the changes from one endpoint to the other. The failure case: if the coordinates are fractional or negative, counting becomes error-prone. For those problems, use the algebra method or the missing endpoint formula instead.

These examples show the method step by step. Each one uses the missing endpoint formula with different coordinate types.

Example 1: Whole Number Coordinates

Endpoint A = (2, 3). Midpoint M = (5, 7). Find endpoint B.

Apply the formula: x₂ = 2M_x - A_x = 2(5) - 2 = 10 - 2 = 8. y₂ = 2M_y - A_y = 2(7) - 3 = 14 - 3 = 11. Missing endpoint B = (8, 11). Check: average of (2,3) and (8,11) is ((2+8)/2, (3+11)/2) = (10/2, 14/2) = (5, 7), which matches M.

Example 2: Negative Coordinates

Endpoint A = (-4, 1). Midpoint M = (0, -3). Find endpoint B.

x₂ = 2M_x - A_x = 2(0) - (-4) = 0 + 4 = 4. y₂ = 2M_y - A_y = 2(-3) - 1 = -6 - 1 = -7. Missing endpoint B = (4, -7). Check: average of (-4,1) and (4,-7) is ((-4+4)/2, (1-7)/2) = (0/2, -6/2) = (0, -3), which matches M. The most common error here is mishandling the subtraction of a negative: 2(0) - (-4) is 0 + 4, not 0 - 4.

Example 3: Fractional Coordinates

Endpoint A = (1.5, 2.25). Midpoint M = (3.75, 5.5). Find endpoint B.

x₂ = 2M_x - A_x = 2(3.75) - 1.5 = 7.5 - 1.5 = 6.0. y₂ = 2M_y - A_y = 2(5.5) - 2.25 = 11.0 - 2.25 = 8.75. Missing endpoint B = (6.0, 8.75). Check: average of (1.5, 2.25) and (6.0, 8.75) is ((1.5+6.0)/2, (2.25+8.75)/2) = (7.5/2, 11.0/2) = (3.75, 5.5), which matches M. For fractional coordinates, an endpoint calculator can help avoid arithmetic mistakes, but the formula is the same.

Three mistakes cause most errors in reverse midpoint problems. Each has a simple fix.

1. Forgetting to multiply the midpoint by 2. The most common error. Students subtract the known endpoint from the midpoint and call the result the missing endpoint. This gives a point that is halfway between the known endpoint and the midpoint, not the other endpoint. Always do 2M - A, not M - A. The midpoint is an average; undoing an average requires multiplication first.

2. Sign errors with negative coordinates. When the known endpoint is negative, subtracting it means adding its absolute value. For example, if A_x = -3 and M_x = 2, then x₂ = 2(2) - (-3) = 4 + 3 = 7. A student who writes 4 - 3 = 1 gets the wrong answer. The fix: always rewrite the subtraction as addition of the opposite value.

3. Confusing the missing endpoint formula with the midpoint formula. The midpoint formula averages two endpoints. The missing endpoint formula reverses that. If you find yourself averaging the midpoint and the known endpoint, you are using the wrong formula. That calculation gives the point at 75% of the segment, not the endpoint. The fix: memorise the missing endpoint formula as 2M - A, not (M + A)/2.

Other mistakes include applying the 2D formula to 3D coordinates (leaving out the z-coordinate) and using an endpoint calculator that expects the inputs in a different order. Always check your answer by computing the midpoint of your result and the known endpoint. If it matches the given midpoint, you are correct.