Question

A map of a state park is drawn so that the origin is placed at the visitor center. The distance between grid lines is 1 mi, Suppose that two hikers are located at points A (-1,3) and B (3,-1) Assume that the units of measurement are in miles,
(a) Determine the distance between the hikers,
(b) 1f the hikers want to meet for lunch, determine the location of the midpoint between the hikers,

Answer 1

To find the distance between the hikers at points A and B and the midpoint between them, we will use the distance formula and the midpoint formula.

### (a) Distance Between the Hikers

The distance \( d \) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in a coordinate plane is given by the formula:

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

For hikers at points \( A(-1, 3) \) and \( B(3, -1) \):

- \( x_1 = -1 \), \( y_1 = 3 \)
- \( x_2 = 3 \), \( y_2 = -1 \)

Plugging these values into the distance formula:

\[
d = \sqrt{(3 - (-1))^2 + (-1 - 3)^2}
\]

Calculating inside the parentheses:

\[
d = \sqrt{(3 + 1)^2 + (-4)^2}
\]
\[
d = \sqrt{(4)^2 + (-4)^2}
\]
\[
d = \sqrt{16 + 16}
\]
\[
d = \sqrt{32}
\]
\[
d = 4\sqrt{2} \approx 5.66 \text{ miles}
\]

### (b) Midpoint Between the Hikers

The midpoint \( M \) between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Using the coordinates of points \( A \) and \( B \):

\[
M = \left( \frac{-1 + 3}{2}, \frac{3 + (-1)}{2} \right)
\]

Calculating the components:

\[
M = \left( \frac{2}{2}, \frac{2}{2} \right)
\]
\[
M = (1, 1)
\]

### Summary

- The distance between the hikers is \( 4\sqrt{2} \) miles (approximately 5.66 miles).
- The midpoint location where the hikers can meet for lunch is at the coordinates \( (1, 1) \).