Question Video: Finding Unknown Coordinates of Points Using the Midpoint Formula Mathematics • 11th Grade

Video Transcript

𝐴, 𝐵, 𝐶, and 𝐷 are collinear points. Suppose that the coordinates of points 𝐴 and 𝐶 are two, four and negative eight, negative eight, respectively, and that the distance between 𝐴 and 𝐵 is equal to the distance between 𝐵 and 𝐶 is equal to the distance between 𝐶 and 𝐷. What are the coordinates of 𝐵 and 𝐷?

In this question, we’re given four collinear points: 𝐴, 𝐵, 𝐶, and 𝐷. And remember when we say collinear points, this means they all lie on the same straight line. We need to determine the coordinates of points 𝐵 and 𝐷. And to do this, we’re given some information. We know that the points lie on the same straight line. We’re given the coordinates of point 𝐴 and the coordinates of point 𝐶. And we’re also given some information about the distances between the points 𝐴, 𝐵, 𝐶, and 𝐷.

Usually, in questions like this, we would start by plotting the information we’re given. However, we don’t yet know the relative positions of the four points, so this would be difficult. Instead, let’s try and determine some information about points 𝐵 and 𝐷. First, we can see that the distance between 𝐴 and 𝐵 is equal to the distance between 𝐵 and 𝐶. In other words, point 𝐵 is equidistant between 𝐴 and 𝐶. And we’re also told that the points are collinear. So, 𝐵 also lies on the line between 𝐴 and 𝐶. And there’s only one point on the line between 𝐴 and 𝐶 equidistant between the two. It’s the midpoint. Therefore, 𝐵 is the midpoint of the line segment 𝐴𝐶.

We can then recall the midpoint of 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two will have coordinates 𝑥 sub one plus 𝑥 sub two over two, 𝑦 sub one plus 𝑦 sub two over two. In other words, we just need to take the average of the 𝑥- and 𝑦-coordinates separately. And we’re given the coordinates of points 𝐴 and 𝐶 in the question. So, we need to substitute 𝑥 sub one is equal to two, 𝑦 sub one is equal to four, 𝑥 sub two is equal to negative eight, and 𝑦 sub two is equal to negative eight into this formula. Doing this gives us the coordinates of 𝐵 are two plus negative eight over two, four plus negative eight over two.

And now, we can evaluate each of the coordinates separately. First, two plus negative eight is negative six, and we divide this by two to get negative three. Next, four plus negative eight is equal to negative four, and we divide this by two to get negative two. Therefore, 𝐵 has coordinates negative three, negative two. And we can actually see something very similar is true for point 𝐷. 𝐷 is also on the same straight line as 𝐴, 𝐵, and 𝐶. And we can see that the distance between 𝐵 and 𝐶 is equal to the distance between 𝐶 and 𝐷. In other words, 𝐶 is the point on the line between 𝐵 and 𝐷 equidistant from both 𝐵 and 𝐷. 𝐶 is the midpoint of the line segment 𝐵𝐷. This means we’ll be able to use our midpoint formula to find an expression for the coordinates of point 𝐶.

First, we’ll start by saying that 𝐷 has coordinates 𝑥, 𝑦. Remember, 𝐷 is the point we want to find. Next, we showed that 𝐵 has coordinates negative three, negative two. Now, since 𝐷 is the midpoint of these two points, we can use our formula to find an expression for the coordinates of 𝐶. However, we already know the coordinates of 𝐶. The 𝑥- and 𝑦-coordinates of 𝐶 are both negative eight. Therefore, the average of the 𝑥-coordinates of points 𝐵 and 𝐷 must be equal to negative eight. Negative eight is equal to 𝑥 plus negative three all over two. Similarly, the average of the 𝑦-coordinates of points 𝐵 and 𝐷 must be the 𝑦-coordinate of point 𝐶. We get negative eight is equal to 𝑦 plus negative two all over two.

We can solve these two equations separately to find the 𝑥- and 𝑦-coordinates of point 𝐷. Let’s start by finding the value of 𝑥. We start by multiplying both sides of our equation through by two and simplify. We get negative 16 is equal to 𝑥 minus three. Then, we add three to both sides of the equation. We see that 𝑥 is equal to negative 13. We can then do the same to find the value of 𝑦. We multiply both sides of the equation through by two and simplify. Negative 16 is equal to 𝑦 minus two. Then, we add two to both sides of the equation. We see that 𝑦 is equal to negative 14, which gives us our final answer.

Therefore, if 𝐴, 𝐵, 𝐶, and 𝐷 are collinear points where point 𝐴 has coordinates two, four and point 𝐶 has coordinates negative eight, negative eight and the distance between 𝐴𝐵 is equal to the distance between 𝐵𝐶 is equal to the distance between 𝐶𝐷, then 𝐵 has coordinates negative three, negative two and 𝐷 has coordinates negative 13, negative 14.