# What is the Endpoint Formula?

Nov 12, 2015

Let's say you had one midpoint given. If you had neither endpoint given nor another midpoint given, then there are an infinite number of endpoints possible and your point is arbitrarily placed (because you only have one point available).

So, to find an endpoint, you need one endpoint and a designated midpoint.

Suppose you have midpoint $M \left(5 , 7\right)$ and the leftmost endpoint $A \left(1 , 2\right)$. That means you have:

${x}_{1} = 1$
${y}_{1} = 2$

So what are $5$ and $7$? The formula for finding the midpoint of a line segment is based on averaging both coordinates in each dimension, assuming 2D cartesian:

$\left(\frac{{x}_{1} + {x}_{\textcolor{red}{2}}}{\textcolor{red}{2}} , \frac{{y}_{1} + {y}_{\textcolor{red}{2}}}{\textcolor{red}{2}}\right)$

where an average is defined as:

$\frac{{a}_{1} + {a}_{2} + {a}_{3} + \ldots + {a}_{\textcolor{red}{N}}}{\textcolor{red}{N}}$

Therefore, you can plug in what you know here to find $B \left({x}_{2} , {y}_{2}\right)$.

$M \left(5 , 7\right) = \left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$

$5 = \frac{{x}_{1} + {x}_{2}}{2} \implies 10 = 1 + {x}_{2}$

$\textcolor{g r e e n}{{x}_{2} = 9}$

$7 = \frac{{y}_{1} + {y}_{2}}{2} \implies 14 = 2 + {y}_{2}$

$\textcolor{g r e e n}{{y}_{2} = 12}$

Therefore, your line segment passes through $A \left(1 , 2\right)$, $M \left(5 , 7\right)$, and $B \left(9 , 12\right)$, and your rightmost endpoint is $\textcolor{b l u e}{B \left(9 , 12\right)}$.