Let's break down the problem step by step:

1. Points and distances:

   • The distance between Point A and Point B is $AB = 250 \, \text{m}$.
   • Point Z is located west of Point B, with a distance $BZ = 350 \, \text{m}$.
   • Point Y is east of Point A, and the distance between $AY$ is the same as $AZ$, i.e., $AY = AZ$.
   • The distance $AZ$ is the sum of $AB + BZ = 250 \, \text{m} + 350 \, \text{m} = 600 \, \text{m}$.
   • Therefore, $AY = 600 \, \text{m}$.

2. Point J:

   • Point J is located directly south of Point Y, and the distance between Point Y and Point J is twice the distance $BY$.
   • We can now calculate $BY$. Since $AY = AZ$, and both $BZ$ and $BY$ are directly along the x-axis (B to Z and B to Y), we calculate $BY = AB = 250 \, \text{m}$.
   • The distance $YJ = 2 \times BY = 2 \times 250 \, \text{m} = 500 \, \text{m}$.

3. Distance between Points Z and J:

   • We now need to determine the distance between Point Z (west of B) and Point J (south of Y). Since Z and J form a right triangle with the legs being:
     • $Z$ to $A$ (horizontal distance): $AZ = 600 \, \text{m}$
     • $J$ to $Y$ (vertical distance): $YJ = 500 \, \text{m}$

4. Using the Pythagorean theorem: $ZJ = \sqrt{(AZ)^2 + (YJ)^2} = \sqrt{600^2 + 500^2} = \sqrt{360000 + 250000} = \sqrt{610000}$ $ZJ = \sqrt{610000} \approx 781.02 \, \text{m}$

Thus, the distance between Point Z and Point J is approximately 781 meters.

Would you like further details or explanations?

Here are 5 related questions:

1. How would the problem change if Point Z were to the north of Point B?
2. What is the significance of the Pythagorean theorem in geometry problems like this?
3. How would the distances change if Point J were to the west of Point Y instead of the south?
4. Can you derive the formula for the distance between two points in 2D space?
5. What if the distance between $Y$ and $J$ was three times $BY$?

Tip: In geometry problems involving distances, drawing a diagram to visualize the points and applying the Pythagorean theorem helps to solve complex setups easily.