Question Video: Finding the Coordinates of Points Using the Midpoint Formula Mathematics • Third Year of Preparatory School

Video Transcript

Consider the points 𝐴: 𝑥, seven; 𝐵: negative four, 𝑦; and 𝐶: two, five. Given that 𝐶 is the midpoint of line segment 𝐴𝐵, find the values of 𝑥 and 𝑦.

First, let’s list out what we know. We have line segment 𝐴𝐵 and 𝐶 is the midpoint point. Point 𝐴 is located at 𝑥, seven. Point 𝐵 is located at negative four, 𝑦. And point 𝐶 is located at two, five. At this point, you might be thinking, “Shouldn’t we try to graph these values?” But because we’re missing this 𝑥- and 𝑦-value, it’s not easy to graph this. So, let’s consider what we know about the midpoint.

The coordinates of the midpoint can be found by averaging the 𝑥-coordinates of the endpoints and the 𝑦-coordinates of the endpoints. And if the midpoint is two, five, then 𝑥 one plus 𝑥 two divided by two has to be equal to two. And 𝑦 one plus 𝑦 two divided by two has to be equal to five. So, we set up two separate equations, one that says 𝑥 one plus 𝑥 two over two equals two and one that says five equals 𝑦 one plus 𝑦 two over two. We’ll let 𝐴 be 𝑥 one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two and then we plug in what we know.

Two is then equal to 𝑥 plus negative four divided by two and five is equal to seven plus 𝑦 divided by two. And now, we just need to solve for each variable. On the left, we multiply both sides of the equation by two, which will give us four equals 𝑥 plus negative four, which we can rewrite to say four equals 𝑥 minus four. Then add four to both sides. And we see that eight equals 𝑥 or, more commonly, 𝑥 equals eight. We follow the same procedure to solve for 𝑦. Multiply by two. Subtract seven from both sides. Three equals 𝑦. So, 𝑦 equals three. Since 𝐴 equals 𝑥 seven and 𝑥 equals eight, point 𝐴 is located at eight, seven. And since 𝐵 was located at negative four, 𝑦 and 𝑦 equals three, 𝐵 is located at negative four, three.