How to Find the Second Endpoint Algebraically when Given One Endpoint and the Midpoint

Last Updated: December 2, 2018

This wikiHow article teaches you how to find the endpoint of a line segment when you're given another endpoint and the midpoint. The formula is as follows: $(x_{3},y_{3})=({\frac {x_{1}+x_{2}}{2}},{\frac {y_{1}+y_{2}}{2}})$ . In this article, the midpoint ( $x_{3},y_{3}$ ) $=$ $(-6,-2)$ and the given endpoint ( $x_{1},y_{1}$ ) $=$ $(-3,-5)$ are used as examples. The rest is algebra: isolate $x_{2}$ and $y_{2}$ (separately) to find the second endpoint coordinates.

Steps

1
Graph your values (optional). This will allow you to get a visual understanding of the linear equation. Make sure that you label your midpoint, $M$ .
2
Rewrite the x coordinate formula, substituting $x_{1}$ and $x_{3}$ .
- For example: ${\frac {-3+x_{2}}{2}}={\frac {-6}{1}}$ ${\frac {-3+x_{{2}}}{2}}={\frac {-6}{1}}$ .
  - $-6={\frac {-6}{1}}$ because they mean the same thing. It's easier to solve when both sides are fractions initially.
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3
Cross multiply to eliminate the denominators. Multiply the top-left and bottom-right side: $(x_{1}+x_{2})\times =2$ $(x_{1}+x_{2})\times =2$ . Repeat with the top-right value by the bottom right value ( $x_{3}$ $x_{{3}}$ and $2$ $2$ ).
- For example:
  - You start with: ${\frac {-3+x_{2}}{2}}={\frac {-6}{1}}$
  - Multiply the top-left and bottom-right: $(-3+x_{2})\times 1=-3+x_{2}$
  - Multiply the bottom-left and top-right (opposite of what was just done): $2\times -6=-12$
  - The result is a simple algebraic equation: $-3+x_{2}=-12$
4
Solve the remaining algebraic equation including isolating the variable. Perform the inverse operation on the constant to isolate the variable—and do it to both sides!
- For example: Perform the inverse operation of $-3$ $-3$ by adding $+3$ $+3$ to the left side. This isolates the variable because the positive reverses the negative. However, you will need to add $+3$ $+3$ to the other side of the equation (beyond the equals sign).
  - $-3{\cancel {+3}}+x_{2}=-12+3$
  - $x_{2}=-9$ . This means that the $x$ value of the second endpoint you're trying to find, is $-9$ .
5
Repeat the process above, now solving for $y$ instead. You've solved half of the problem by finding the $x$ $x$ coordinate of the endpoint. Now, your endpoint is $(-9,y)$ $(-9,y)$ . As you can see, you're not done! Here's how to solve for $y$ $y$ :
- Substitute $y_{1}$ and $y_{3}$ : ${\frac {-5+y_{2}}{2}}={\frac {-2}{1}}$
- Cross multiply and isolate the variable: $-5{\cancel {+5}}+y_{2}=-4+5$
- Y coordinate: $y_{2}=1$
- Final answer (missing endpoint): $(-9,1)$