The Frostburg-Truth bus travels from Frostburg Mall through the city’s center to Sojourner Truth Park. The mall is 3 miles west and 4 miles south of the city’s center. Truth Park is 3 miles east and 2 miles north of the city’s center. How far is it from Truth Park to the mall to the nearest tenth of a mile?

To find the distance between Truth Park and the mall, we can use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is given by:

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

### Step 1: Determine the coordinates of the mall and Truth Park relative to the city center.

- The mall is 3 miles west and 4 miles south of the city center, so its coordinates relative to the center are \((-3, -4)\).
- Truth Park is 3 miles east and 2 miles north of the city center, so its coordinates relative to the center are \((3, 2)\).

### Step 2: Apply the distance formula.

Using the coordinates for the mall \((-3, -4)\) and Truth Park \((3, 2)\), we calculate:

\[
d = \sqrt{(3 - (-3))^2 + (2 - (-4))^2}
\]

Simplify inside the parentheses:

\[
d = \sqrt{(3 + 3)^2 + (2 + 4)^2}
\]

\[
d = \sqrt{6^2 + 6^2}
\]

\[
d = \sqrt{36 + 36}
\]

\[
d = \sqrt{72}
\]

### Step 3: Simplify and find the distance.

\[
d = \sqrt{72} \approx 8.5 \text{ miles}
\]

So, the distance from Truth Park to the mall is approximately \(\mathbf{8.5}\) miles.

Would you like further explanation or have any questions?

Here are some related questions:
1. How do you calculate the distance between two points in three-dimensional space?
2. What is the significance of the distance formula in real-world applications?
3. How would you find the midpoint of the segment connecting the mall and Truth Park?
4. How can you determine the coordinates of a point that divides a segment in a given ratio?
5. How would the distance change if the coordinates were given on a different scale?

**Tip:** When dealing with distances, always double-check your signs and operations inside the square root to ensure accuracy.

Learn how to calculate the distance between Truth Park and the mall using coordinate geometry and the distance formula. This problem involves determining coordinates relative to a city center and applying the distance formula to find the exact distance in miles. Suitable for Grades 7-9 students.

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