Missing Endpoint From Midpoint

Missing Endpoint from a Midpoint

When the midpoint of a segment and one endpoint are known, you can find the missing endpoint by working backward from the midpoint formula.

1) Use the midpoint idea

For a segment with endpoints ((x_1, y_1)) and ((x_2, y_2)), the midpoint is halfway between them in both coordinates. That means the midpoint’s x-value is the average of the endpoints’ x-values, and the same is true for the y-values.

2) Solve one coordinate at a time

If the midpoint is ((m_x, m_y)) and one endpoint is ((x_1, y_1)), let the missing endpoint be ((x_2, y_2)).

Set up two equations:

• (\frac{x_1 + x_2}{2} = m_x)
• (\frac{y_1 + y_2}{2} = m_y)

Then solve each equation for the unknown coordinate.

A quick way is to multiply the midpoint coordinate by 2, then subtract the known endpoint coordinate:

• (x_2 = 2m_x - x_1)
• (y_2 = 2m_y - y_1)

3) Simplify carefully

Carry out the arithmetic with signs, fractions, or decimals as needed. Write the ordered pair in simplest form.

4) Check your answer

Verify that averaging the two endpoints gives the stated midpoint. If both coordinates match, the missing endpoint is correct.