Video: Finding a Side Length in a Triangle Using Properties of Medians

Video Transcript

In triangle 𝐽𝐾𝐿, 𝐽𝑃 equals six centimeters. Find the length of line segment 𝑃𝑆.

We need to know the length of line segment 𝑃𝑆, and we’ve been given that 𝐽𝑃 equals six centimeters. First of all, we should note that the point 𝑆 divides the line segment 𝐾𝐿 in half, which makes 𝑆 a midpoint. Since 𝐽 is a vertex of this triangle, we know that the distance between a vertex and the midpoint is called the median. And so we can say that 𝐽𝑆 is a median. In the same way, 𝑇 divides line segment 𝐽𝐿 and 𝑅 divides line segment 𝐽𝐾, which means 𝐾𝑇 is a median and 𝐿𝑅 is a median. Since 𝐽𝑆, 𝐾𝑇, and 𝐿𝑅 are all meet at one point 𝑃, 𝑃 is the point of concurrency or the centroid.

This is important because we know something about the centroid. For a median, the distance between the vertex and the centroid is two-thirds of the median. And the distance between the centroid and the midpoint is one-third of the distance of that median. To go from two-thirds to one-third, we multiply by one-half. One-third is half of two-thirds. And so we can say that the line segment 𝑃𝑆 will be equal to one-half the line segment 𝐽𝑃. Since 𝐽𝑃 was equal to six centimeters, we take one half of that and we say that the line segment 𝑃𝑆 is equal to three centimeters.