Final answer:
To find the length of CK, we first determine the value of t by setting CP equal to PK since RP is a perpendicular bisector. After calculating t to be 12, we then find CP (equivalent to PK) to be 70. CK, being the sum of CP and PK, equals 140.
Step-by-step explanation:
If RP is a perpendicular bisector, it means that RP cuts a line segment into two equal parts and it is at a right angle to it. Here, since CP = PK (because RP is a perpendicular bisector), we can set CP equal to PK and solve for the variable t. The lengths given are CP = 5t + 10 and PK = 4t + 22.
Setting CP equal to PK we get:
5t + 10 = 4t + 22
Solving for t gives us:
5t - 4t = 22 - 10
t = 12
Since CP = 5t + 10, we can now find the length of CP (which is equal to the length of PK because RP is a perpendicular bisector):
CP = 5(12) + 10 = 60 + 10
CP = 70
Therefore, the length of CK (which is CP + PK) is 70 + 70, as CP equals PK.
CK = 140