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