Final answer:
The value of x is 11. By setting the radius PC equal to the other two radii to the vertices (RC and QC) and solving these equations, we can determine that x equals 11.
Step-by-step explanation:
The student has provided the lengths of the radii to the three vertices of a triangle, namely PC, RC, and QC. Since C is the circumcenter of triangle PQR, all the radii to the vertices (PC, RC, and QC) are equal. We can set up an equation for each pair of radii.
First, set PC equal to RC:
3x + 7 = 5x - 15
Solve for x:
7 + 15 = 5x - 3x
22 = 2x
x = 11
Next, set PC equal to QC:
3x + 7 = 51 - x
Again, solve for x:
3x + x = 51 - 7
4x = 44
x = 11
Since both equations give us the same value for x, we can confirm that x = 11.