Answer
$QB = 9$
$QP = 3$
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 the one part coming from the vertex is two-thirds of the median and the other part attaching to the side opposite the vertex makes up one-third of the median.
We can then say that $PB$ is two-thirds of $QB$ and $QP$ is one-third of $BP$. Let's set up an equation to find $QB$:
$PB = \frac{2}{3}(QB)$
Let's plug in what we know:
$6 = \frac{2}{3}(QB)$
Let's divide each side by $\frac{2}{3}$, meaning we multiply by its reciprocal, $\frac{3}{2}$:
$QB = 6(\frac{3}{2})$
Multiply to solve:
$QB = \frac{18}{2}$
Divide both the numerator and denominator by their greatest common factor, $2$:
$QB = 9$
We know that $QP$ is one-third of $QB$, so let's set up the equation to find $QP$:
$QP = \frac{1}{3}(9)$
Multiply:
$QP = \frac{9}{3}$
Divide the numerator and denominator by their greatest common factor, $3$:
$QP = 3$
