Line segment 𝐴𝐷 is a median in triangle 𝐴𝐵𝐶, where 𝐴 equals eight, negative seven and 𝐷 equals two, negative one. Find the point of intersection of the medians of the triangle 𝐴𝐵𝐶.
Let’s say that this is that triangle with corners 𝐴, 𝐵, 𝐶. And we’re told about a line segment 𝐴𝐷, which is a median of this triangle. Point 𝐷 then we know is midway between points 𝐵 and 𝐶, and so the line segment 𝐴𝐷 looks like this. We’re given the coordinates of points 𝐴 and 𝐷, and we want to solve for the point of intersection of the medians of this triangle. The other two medians would look a bit like this. So the point of intersection of all three is here.
For any triangle, this point of intersection of the medians is located two-thirds of the way along any one of the medians. That is, for the median we’re interested in, line segment 𝐴𝐷, if we divide the length of that segment into three equal parts, two of those parts span from point 𝐴 to the intersection point and the remaining part spans from the intersection point to point 𝐷. We can say then that the point of intersection of the medians is two-thirds of the way along line segment 𝐴𝐷.
What we want to do is find the coordinates of this point right here. We’ll call it point 𝑃. To start doing that, we can recall that if we have a line segment between two points whose coordinates are known to us and we want to solve for the coordinates of a point that divides that line segment up into a ratio that is known, then the coordinates of point 𝑃 are given by this expression. Here, we’re using the ratio into which the line is divided up, 𝑚 and 𝑛, as well as the 𝑥-coordinates of our second and first point and the 𝑦-coordinates of those respective points.
If we apply this relationship to our scenario, to solve for the coordinates of the point of intersection of these medians, then we could say that 𝑚 and 𝑛, the ratio into which our line is divided, are two and one. And the coordinates 𝑥 one and 𝑦 one are those of point 𝐴, while 𝑥 two and 𝑦 two are those of point 𝐷. Carefully substituting all these values in, we get this expression for the coordinates of our point 𝑃. Simplifying these fractions, we have 12 over three and negative nine over three. This simplifies further to four and negative three. These, then, are the coordinates of the point of intersection of the medians of the triangle 𝐴𝐵𝐶.