A point P moves so that the sum of its distances from the lines x = 1 and y = -2 is always 8…

Step 1. The rule. $|x - 1| + |y + 2| = 8$. With $u = x - 1$ and $v = y + 2$ this is $|u| + |v| = 8$.

Step 2. The shape. In the $(u, v)$ plane the four cases give the sides of a square with corners $(\pm 8, 0)$ and $(0, \pm 8)$. In the $(x, y)$ plane the corners are $(9, -2)$, $(-7, -2)$, $(1, 6)$ and $(1, -10)$.

Step 3. The area. Both diagonals are 16 long, so the area is $\dfrac{16 \times 16}{2} = 128$.

Step 4. The perimeter. One side joins $(9, -2)$ to $(1, 6)$, so it is $\sqrt{8^2 + 8^2} = 8\sqrt{2}$. The perimeter is $4 \times 8\sqrt{2} = 32\sqrt{2} \approx 45.25$.