Answer: PC = 5
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Work Shown:
AB+BC+CD = AD .... segment addition postulate
BC+BC+CD = AD ... replace AB with BC (since AB = BC)
BC+BC+BC = AD ... replace CD with BC (since BC = BC)
3*BC = AD .... combine like terms
3*BC = 12 .... replace AD with 12
BC = 12/3 .... divide both sides by 3
BC = 4
For right triangle PBC, the legs are BP = 3 and BC = 4. We can use the Pythagorean Theorem to find that the hypotenuse is PC = 5
The steps are shown below
(BP)^2+(BC)^2 = (PC)^2
3^2+4^2 = (PC)^2
9+16 = (PC)^2
25 = (PC)^2
(PC)^2 = 25
PC = sqrt(25)
PC = 5
The other parts of the diagram seem to be thrown in as a distraction.