#### Answer

$AR = 18$
$AP = 12$

#### Work Step by Step

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, $3$:
$AP = 12$