Question

Grantville Town Park is located at the point (0,0). The Riley family lives 6 miles west and 3 miles north of the park. The Brown family lives 2 miles east and 4 miles south from the park. What is the shortest distance between their houses? Round your answer to the nearest tenth.

Answer 1

To find the coordinates of the houses of the Riley and Brown families, we can start from the given coordinates of Grantville Town Park at \((0,0)\).

**Riley family coordinates:**
- They live 6 miles west and 3 miles north of the park.
- West means we move left on the x-axis, so we subtract 6 from the x-coordinate.
- North means we move up on the y-axis, so we add 3 to the y-coordinate.
- Thus, the coordinates for the Riley family are:
\[
(-6, 3)
\]

**Brown family coordinates:**
- They live 2 miles east and 4 miles south from the park.
- East means we move right on the x-axis, so we add 2 to the x-coordinate.
- South means we move down on the y-axis, so we subtract 4 from the y-coordinate.
- Thus, the coordinates for the Brown family are:
\[
(2, -4)
\]

Now that we have the coordinates of both families:
- Riley family: \((-6, 3)\)
- Brown family: \((2, -4)\)

Next, we can calculate the shortest distance between these two points using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1) = (-6, 3)\) and \((x_2, y_2) = (2, -4)\).

Substituting the values into the distance formula:
\[
d = \sqrt{(2 - (-6))^2 + (-4 - 3)^2}
\]
\[
= \sqrt{(2 + 6)^2 + (-4 - 3)^2}
\]
\[
= \sqrt{(8)^2 + (-7)^2}
\]
\[
= \sqrt{64 + 49}
\]
\[
= \sqrt{113}
\]
To find the numerical value, we calculate:
\[
\sqrt{113} \approx 10.63014581273465
\]

Rounding this to the nearest tenth:
\[
\approx 10.6
\]

Thus, the shortest distance between the houses of the Riley and Brown families is \(\boxed{10.6}\) miles.