# A segment has a midpoint (-2, 9) and one endpoint (2,8). What is the coordinate of the other endpoint?

Jun 7, 2017

See a solution process below:

#### Explanation:

The formula to find the mid-point of a line segment give the two end points is:

$M = \left(\frac{\textcolor{red}{{x}_{1}} + \textcolor{b l u e}{{x}_{2}}}{2} , \frac{\textcolor{red}{{y}_{1}} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

Where $M$ is the midpoint and the given points are:

$\left(\textcolor{red}{{x}_{1} , {y}_{1}}\right)$ and $\left(\textcolor{b l u e}{{x}_{2} , {y}_{2}}\right)$

Subtituting gives:

$\left(- 2 , 9\right) = \left(\frac{\textcolor{red}{2} + \textcolor{b l u e}{{x}_{2}}}{2} , \frac{\textcolor{red}{8} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

Solving for $x$ gives:

$- 2 = \frac{\textcolor{red}{2} + \textcolor{b l u e}{{x}_{2}}}{2}$

$\textcolor{g r e e n}{2} \times - 2 = \textcolor{g r e e n}{2} \times \frac{\textcolor{red}{2} + \textcolor{b l u e}{{x}_{2}}}{2}$

$- 4 = \cancel{\textcolor{g r e e n}{2}} \times \frac{\textcolor{red}{2} + \textcolor{b l u e}{{x}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}}$

$- 4 = \textcolor{red}{2} + \textcolor{b l u e}{{x}_{2}}$

$- 2 - 4 = - 2 + \textcolor{red}{2} + \textcolor{b l u e}{{x}_{2}}$

$- 6 = 0 + \textcolor{b l u e}{{x}_{2}}$

$- 6 = \textcolor{b l u e}{{x}_{2}}$

$\textcolor{b l u e}{{x}_{2}} = - 6$

Solving for $y$ gives:

$9 = \frac{\textcolor{red}{8} + \textcolor{b l u e}{{y}_{2}}}{2}$

$\textcolor{g r e e n}{2} \times 9 = \textcolor{g r e e n}{2} \times \frac{\textcolor{red}{8} + \textcolor{b l u e}{{y}_{2}}}{2}$

$18 = \cancel{\textcolor{g r e e n}{2}} \times \frac{\textcolor{red}{8} + \textcolor{b l u e}{{y}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}}$

$18 = \textcolor{red}{8} + \textcolor{b l u e}{{y}_{2}}$

$- 8 + 18 = - 8 + \textcolor{red}{8} + \textcolor{b l u e}{{y}_{2}}$

$10 = 0 + \textcolor{b l u e}{{y}_{2}}$

$10 = \textcolor{b l u e}{{y}_{2}}$

$\textcolor{b l u e}{{y}_{2}} = 10$

The other end point is: $\left(\textcolor{b l u e}{- 6 , 10}\right)$

Jun 7, 2017

The coordinate of the other endpoint is $\left(- 6 , 10\right)$

#### Explanation: This is a simple number line from $0$ to $10$. Now, I ask you, what is an easy way to calculate the midpoint between $0$ and $10$? Of course it's $5$ but is there a formula we can derive to calculate the midpoint between any two points? Well, let's see. If we took $0$ and $10$, added them together, then divided the quotient by $2$, boom! We get $5$!

Let's try it with two more numbers. How about $2$ and $4$? If we were to add the two up, then divide the result by $2$, we'd get $3$ - also the midpoint between $2$ and $4$.

Now we can see that there's a formula that we can use to calculate the midpoint between any two numbers! Add the two end numbers up and divide the result by $2$! This formula - known as the Midpoint Formula - is shown below:

For any two endpoints ${x}_{1}$ and ${x}_{2}$ on the number line,

$M = \frac{{x}_{1} + {x}_{2}}{2}$

So how does all of this relate to the problem? Well, remember that the graph is a coordinate plane, made up of two axes - the x-axis and the y-axis. And you can think of each axis to be a number line!

So now, our task is to derive a formula for finding the midpoint between any two points on the coordinate point. That way, we can write a relationship between the two endpoints on the coordinate plane and the midpoint between those two points.

Suppose there was a graph like the one shown below: Suppose point A had the coordinates $\left({x}_{1} , {y}_{1}\right)$ and point B had the coordinates $\left({x}_{2} , {y}_{2}\right)$. And suppose there was also a point M that was the midpoint between points A and B. Now, we want to write the midpoint's coordinates in terms of the $x$ and $y$ coordinates of points A and B. That way, we can connect the coordinates of points A and B with the coordinates of their midpoint. Now, the coordinates of the midpoint are essentially the midpoint between the two x-coordinates and the midpoint between the two y-coordinates.

Let's focus on the x-axis first and treat it as a number line, except without numbers. On the x-axis, we have the x-coordinate of point A as well as the x-coordinate of point B. From our earlier rule, we can state that the midpoint between the two x-coordinates is:

$M = \frac{{x}_{1} + {x}_{2}}{2}$

Likewise, the midpoint between the two y-coordinates is:

$M = \frac{{y}_{1} + {y}_{2}}{2}$

As we stated earlier, the coordinates of the midpoint between points A and B are the midpoint between the two x-coordinates and the midpoint between the two y-coordinates. Combining the two statements above, we get the conclusion that the coordinates of the midpoint between points A and B are:

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

Now, with that information, we can substitute in the values mentioned in the question. We let point A and its coordinates be our missing endpoint, point B and its coordinates be our known endpoint, and point M, our midpoint, and its coordinates be the midpoint, like so:

$M \left(- 2 , 9\right) = \left(\frac{{x}_{1} + 2}{2} , \frac{{y}_{1} + 8}{2}\right)$

This equation tells us that $- 2 = \frac{{x}_{1} + 2}{2}$ and that $9 = \frac{{y}_{1} + 8}{2}$. Next, we start solving the equations to figure out what ${x}_{1}$ and ${y}_{1}$ are.

$- 2 = \frac{{x}_{1} + 2}{2}$

${x}_{1} + 2 = - 4$

${x}_{1} = - 6$

And now, let's solve the right equation:

$9 = \frac{{y}_{1} + 8}{2}$

${y}_{1} + 8 = 18$

${y}_{1} = 10$

And now, we put the two values together to form the answer:

$\left(- 6 , 10\right)$