Answer:
The midpoint of the line segment is located at (-4, 4).
Step-by-step explanation:
We're given the coordinate points of a line that can help us find the midpoint.
The midpoint formula for a line is written as:
![\bullet \ \ \ \displaystyle\big(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\big)](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5C%20%5C%20%5Cdisplaystyle%5Cbig%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%20%5Cfrac%7By_1%2By_2%7D%7B2%7D%5Cbig%29)
Additionally, we are given the coordinate points (5, -4) and (-13, 12). We can use these and label them with the (x, y) system so we can substitute them into the formula.
In math, a coordinate pair is written as (x, y). This is where cos = x and sin = y. If we are given two coordinate pairs, we can label them with the (x, y) system but also incorporating a subscript to distinguish the two x-values from each other as well as the y-values. We do this by turning the two x-values into x₁ and x₂ and the y-values follow the same protocol: y₁ and y₂.
Therefore, we can label our two coordinates:
<u>(5, -4)</u>
<u>(-13, 12)</u>
Now, we can place these values into the midpoint formula and simplify to find our midpoint.
Recall that the midpoint formula is:
![\bullet \ \ \ \displaystyle\big(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\big)](https://tex.z-dn.net/?f=%5Cbullet%20%5C%20%5C%20%5C%20%5Cdisplaystyle%5Cbig%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%20%5Cfrac%7By_1%2By_2%7D%7B2%7D%5Cbig%29)
Therefore, let's substitute these values.
![\displaystyle\big(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\big)\\\\\\\big(\frac{5 + (-13)}{2}, \frac{(-4)+12}{2}\big)\\\\\\\big(\frac{-8}{2}, \frac{8}{2}\big)\\\\\\\boxed{(-4, 4)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cbig%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%20%5Cfrac%7By_1%2By_2%7D%7B2%7D%5Cbig%29%5C%5C%5C%5C%5C%5C%5Cbig%28%5Cfrac%7B5%20%2B%20%28-13%29%7D%7B2%7D%2C%20%5Cfrac%7B%28-4%29%2B12%7D%7B2%7D%5Cbig%29%5C%5C%5C%5C%5C%5C%5Cbig%28%5Cfrac%7B-8%7D%7B2%7D%2C%20%5Cfrac%7B8%7D%7B2%7D%5Cbig%29%5C%5C%5C%5C%5C%5C%5Cboxed%7B%28-4%2C%204%29%7D)
Therefore, the midpoint of the line segment is located at (-4, 4), which is Option A.