- Identify the coordinates of the two points: \(P = (-2, 3)\) and \(N = (3, -1)\).
- Apply the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
- Substitute the coordinates: \(d = \sqrt{(3 - (-2))^2 + (-1 - 3)^2} = \sqrt{5^2 + (-4)^2}\).
- Calculate the square root: \(\sqrt{25 + 16} = \sqrt{41}\).
- Select the matching option: \(\boxed{\sqrt{41}}\).
Explanation
- Identify the coordinates and the distance formula.
The distance between two points \(P(x_1, y_1)\) and \(N(x_2, y_2)\) on a coordinate plane can be determined using the distance formula, which is derived from the Pythagorean theorem:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
In this problem, we are given:
- Pablo's location: \(P = (-2, 3)\), so \(x_1 = -2\) and \(y_1 = 3\).
- Nadia's location: \(N = (3, -1)\), so \(x_2 = 3\) and \(y_2 = -1\).
- Calculate the distance using the coordinate values.
Now, substitute these coordinate values into the distance formula:
\[d = \sqrt{(3 - (-2))^2 + (-1 - 3)^2}\]
Next, simplify the expressions inside the parentheses:
- \(3 - (-2) = 3 + 2 = 5\)
- \(-1 - 3 = -4\)
Substituting these values back:
\[d = \sqrt{(5)^2 + (-4)^2}\]
\[d = \sqrt{25 + 16}\]
\[d = \sqrt{41}\]
- Conclusion and final answer.
The calculated distance between the two playing pieces is \(\sqrt{41}\). Comparing this to the given options:
A\(\sqrt{45}\)
B\(\sqrt{22}\)
C\(\sqrt{41}\)
D\(\sqrt{29}\)
The result corresponds to option (c). The final answer is \(\boxed{\sqrt{41}}\).
Examples
The distance formula is a fundamental tool for finding the length of a segment between two locations on a map or grid. For instance, if you are planning a route between two cities represented as points on a grid, you can use this formula to calculate the straight-line distance, helping you estimate travel times and fuel requirements. In our case, the distance between the two pieces is \(\sqrt{41} \approx 6.4\) units.