Question

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

Answer 1

See a solution process below:

Explanation:

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

`M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)`

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

`(color(red)(x_1, y_1))` and `(color(blue)(x_2, y_2))`

Subtituting gives:

`(-2, 9) = ((color(red)(2) + color(blue)(x_2))/2 , (color(red)(8) + color(blue)(y_2))/2)`

Solving for `x` gives:

`-2 = (color(red)(2) + color(blue)(x_2))/2`

`color(green)(2) xx -2 = color(green)(2) xx (color(red)(2) + color(blue)(x_2))/2`

`-4 = cancel(color(green)(2)) xx (color(red)(2) + color(blue)(x_2))/color(green)(cancel(color(black)(2)))`

`-4 = color(red)(2) + color(blue)(x_2)`

`-2 - 4 = -2 + color(red)(2) + color(blue)(x_2)`

`-6 = 0 + color(blue)(x_2)`

`-6 = color(blue)(x_2)`

`color(blue)(x_2) = -6`

Solving for `y` gives:

`9 = (color(red)(8) + color(blue)(y_2))/2`

`color(green)(2) xx 9 = color(green)(2) xx (color(red)(8) + color(blue)(y_2))/2`

`18 = cancel(color(green)(2)) xx (color(red)(8) + color(blue)(y_2))/color(green)(cancel(color(black)(2)))`

`18 = color(red)(8) + color(blue)(y_2)`

`-8 + 18 = -8 + color(red)(8) + color(blue)(y_2)`

`10 = 0 + color(blue)(y_2)`

`10 = color(blue)(y_2)`

`color(blue)(y_2) = 10`

The other end point is: `(color(blue)(-6, 10))`

Answer 2

The coordinate of the other endpoint is `(-6, 10)`

Explanation:

Instead of full-on visualizing a graph, let's start with a number line:

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=(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 `(x_1, y_1)` and point B had the coordinates `(x_2, y_2)`. 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=(x_1+x_2)/2`

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

`M=(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=((x_1+x_2)/2, (y_1+y_2)/2)`

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(-2, 9)=((x_1+2)/2, (y_1+8)/2)`

This equation tells us that `-2=(x_1+2)/2` and that `9=(y_1+8)/2`. Next, we start solving the equations to figure out what `x_1` and `y_1` are.

Let's start with the left equation:

`-2=(x_1+2)/2`

`x_1+2=-4`

`x_1=-6`

And now, let's solve the right equation:

`9=(y_1+8)/2`

`y_1+8=18`

`y_1=10`

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

`(-6, 10)`