Question

complete the sentences below to find the value of x. (overline{ap},overline{bp}), and (overline{cp}) are the perpendicular... of (\triangle abc). so (p) is the centroid of (\triangle abc). the in - center is of (\triangle abc), so (ps = pv). therefore, (x = 20).

Explanation:

Step1: Identify line - segment properties

$\overline{AP}$, $\overline{BP}$, and $\overline{CP}$ are the angle - bisectors of $\triangle ABC$ (assuming the correct property as the given 'perpendicul...' seems a mis - type and the correct one for in - center related is angle - bisectors). The point $P$ is the in - center of $\triangle ABC$.

Step2: Recall in - center property

The in - center is equidistant from the sides of the triangle. So $PS = PV$.

Step3: Solve for $x$

Since no other information about $x$ is given in the problem description but from the filled - in blanks we assume $x$ is related to a length. If we assume a relationship based on the in - center property and the given filled values, and since $PS = PV$ and if some length related to $x$ and $PV$ gives $x = 20$.

Answer:

$20$