From a point P, R is 5km due West and 12km due South. Find the distance between P and R'.

To find the distance between P and R', we first need to determine the location of R'. R' is the reflection of R across the point P. Since R is 5km due West and 12km due South of P, we can draw a diagram to represent their positions.

           R
            |
            | 12km
            |
            P------5km------->
            |
            |

To find R', we draw a line that connects R and P, then draw a perpendicular bisector to that line at point P. The intersection of the perpendicular bisector and the line RP is the point R'.

           R'
            |
            | 
            |
            P------5km------->
            |
            |

The distance between P and R' is the length of the line segment PR'. To find this distance, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, we have a right triangle with sides of length 5km and 12km. The hypotenuse, which is the distance between P and R', is given by:

sqrt(5^2 + 12^2) = sqrt(169) = 13km

Therefore, the distance between P and R' is 13km.