Liam plotted the locations of several places in his town in this coordinate plane. Liam’s house lies halfway between the museum and the movie theater. Find the coordinates at which Liam should graph a point representing his house.
In this question, we’re told that Liam plots the locations of several places in his town in the coordinate plane. And we’re given a diagram with this information. We’re then told that Liam’s house lies halfway between the museum and the movie theater. We need to use this information to determine the coordinates we should plot Liam’s house on this coordinate plane. To do this, we can start by recalling what it means to say a point is halfway between two other points in the coordinate plane. The halfway point will be the point on the line segment which is equidistant from the endpoints. This is often called the midpoint.
Another way of saying this is the midpoint of a line segment from 𝐴 to 𝐵 is the point 𝐶 on the line segment 𝐴𝐵 such that the distance from 𝐴 to 𝐶 is equal to the distance from 𝐶 to 𝐵. And this actually gives us two ways of answering this question. We’ll go through both of these.
First, let’s start by noting where the movie theater and the museum lie in our coordinate plane. Since we want to find the halfway or midpoint of these two points, let’s draw the line segment connecting them. Now, the halfway point will be halfway between these two points. It’s equidistant from them both. And we can see in the diagram there’s 12 units’ difference between the two. Therefore, for a point on this line to be equidistant from both, it must be six units away from both. This means we just count six units from either the movie theater or the museum to find the midpoint. Then, all we need to do is read off the coordinates of this point. It’s five units across and negative two units down. So the coordinates of the house will be five, negative two.
However, this method required us to have a sketch of the diagram. We can also answer this question by using only the coordinates of the endpoints of this line segment. To do this, we recall we can find the midpoint of the two points 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two by finding the average of the 𝑥-coordinates and 𝑦-coordinates. The midpoint will have coordinates 𝑥 sub one plus 𝑥 sub two all over two, 𝑦 sub one plus 𝑦 sub two all over two. We can use this formula to determine the coordinates of the house. We first need to find the coordinates of the movie theater and the museum.
Let’s start by finding the coordinates of the movie theater from the diagram. It’s five across and four up, so it has coordinates five, four. We can do the same to find the coordinates of the museum. It’s five units to the right and eight units down, so it has coordinates five, negative eight.
We now need to substitute the coordinates of these two points into our formula for the midpoint. 𝑥 sub one is five, 𝑦 sub one is four, 𝑥 sub two is five, and 𝑦 sub two is negative eight. Let’s start by finding the 𝑥-coordinate of the midpoint. We get five plus five all over two, which simplifies to 10 over two, which we can evaluate is equal to five. We can do the same for the 𝑦-coordinate of the midpoint. We substitute 𝑦 sub one is four and 𝑦 sub two is negative eight to get four plus negative eight all over two. And of course four plus negative eight is equal to negative four. And we divide this by two to get negative two. And this of course gives us the same answer.
Therefore, we were able to show in the coordinate plane Liam’s house will have coordinates five, negative two.