If someone lives 6 miles west and 3 miles north and another person lives 2 miles east and 4 miles south how far apart do they live?

First, let's translate the given directions into coordinates. 

- The first person lives 6 miles west and 3 miles north. We can represent this as the point \((-6, 3)\).
- The second person lives 2 miles east and 4 miles south. We can represent this as the point \((2, -4)\).

Now, we'll use the distance formula:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting in the coordinates:

\[
\text{Distance} = \sqrt{(2 - (-6))^2 + (-4 - 3)^2}
\]

\[
\text{Distance} = \sqrt{(2 + 6)^2 + (-4 - 3)^2}
\]

\[
\text{Distance} = \sqrt{(8)^2 + (-7)^2}
\]

\[
\text{Distance} = \sqrt{64 + 49}
\]

\[
\text{Distance} = \sqrt{113} \approx 10.63
\]

So, the two people live approximately **10.63 miles** apart.

Do you want more details or have any questions?  
Here are some related questions to explore:

1. How can we determine the direction from one person to the other?
2. How would the distance change if the second person was 4 miles east and 2 miles south?
3. How can you represent this scenario graphically on a coordinate plane?
4. What if both people lived in a 3D space—how would the distance formula change?
5. How would you calculate the midpoint between their locations?

**Tip:** When dealing with directions like "west" or "north," treat them as negative or positive movements along the coordinate axes.

Learn how to calculate the distance between two points given in a coordinate plane scenario. This problem involves using the distance formula to find the distance between two locations: one 6 miles west and 3 miles north, and another 2 miles east and 4 miles south. Suitable for students in Grades 6-8.

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