Geometry: Common Core (15th Edition)

$AR = 18$ $AP = 12$
A centroid is the point where all the medians of a triangle meet. The median is said to be concurrent at point $P$. The centroid also divides the median into two parts where one part is two-thirds of the median and the other part makes up one-third. We can then say that $AP$ is two-thirds of $AR$ and $PR$ is one-third of $AR$. Let's set up an equation to find $AP$: $PR = \frac{1}{3}(AR)$ Let's plug in what we know: $6 = \frac{1}{3}(AR)$ Let's divide each side by $\frac{1}{3}$, meaning we multiply by its reciprocal, $3$: $AR = 6(3)$ Multiply to solve: $AR = 18$ We know that $AP$ is two-thirds of $AR$. Let's set up the equation to find $AP$: $AP = \frac{2}{3}(18)$ Multiply: $AP = \frac{36}{3}$ Divide the numerator and denominator by their greatest common factor, $3$: $AP = 12$