Final answer:
Using the properties of the circumcenter in a triangle, where the circumcenter is equidistant from all vertices, we set the equations for PC and RC (or QC) equal to each other, solved for X, and found that PC equals 5 units.
Step-by-step explanation:
In the context of geometry, the circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect, and it is equidistant from the vertices of the triangle.
If C is the circumcenter of triangle PQR, then PC, RC, and QC are all radii of the circumcircle and thus should be equal to each other. Given the equations PC = 3X - 7, RC = 5X - 15, and QC = 51 - X, we can find the value of X by setting PC equal to RC or QC since they all represent the same length due to the properties of the circumcenter.
Setting PC equal to RC:
3X - 7 = 5X - 15
15 - 7 = 5X - 3X
8 = 2X
X = 4
Now, substitute X back into the equation for PC to get:
PC = 3(4) - 7
PC = 12 - 7
PC = 5
Therefore, the length of PC is 5 units.