Find the length of line segment
𝐴𝑀, given that 𝐴𝐸 equals 54.
Let’s see what we can tell from the
diagram. The point 𝐷 divides line segment
𝐴𝐵 in half, and the point 𝐸 divides line segment 𝐵𝐸 in half. So we have two midpoints. And we know that 𝐴 and 𝐶 are
vertices of this triangle, which means that line segment 𝐴𝐸 and line segment 𝐶𝐷
are medians of this triangle. The place where medians intersect
inside a triangle is called the point of concurrency, or the centroid. And we know based on the centroid
theorem that the distance from the vertex to the centroid is two-thirds of the
median, and the distance from the centroid to the midpoint is one-third of the
This means line segment 𝐴𝑀 is
equal to two-thirds the median 𝐴𝐸. And since we know 𝐴𝐸, the median,
equals 54, we can say that the length of line segment 𝐴𝑀 will be equal to
two-thirds of 54. If we wanna simplify this, I know
that 54 divided by three equals 18 and two times 18 equals 36. So we can say that line segment
𝐴𝑀 measures 36.