Question

The distance between the points \((-2, 5)\) and \((5, -2)\) is

• A. \(7\sqrt{2}\)
• B. 14
• C. \(2\sqrt{7}\)
• D. 7

Hint:

When the absolute differences of both coordinates are equal, i.e., \(|x_2 - x_1| = |y_2 - y_1| = a\), the distance is always \(a\sqrt{2}\).
Here, the difference is \(|5 - (-2)| = 7\) and \(|-2 - 5| = 7\), so the distance is immediately \(7\sqrt{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are asked to find the distance between two points in a two-dimensional Cartesian coordinate system.
The given coordinates are \(P(x_1, y_1) = (-2, 5)\) and \(Q(x_2, y_2) = (5, -2)\).

Step 2: Key Formula or Approach:
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Detailed Explanation:
1. Let the given points be: \[ (x_1, y_1) = (-2, 5) \] \[ (x_2, y_2) = (5, -2) \] 2. Substitute these coordinates into the distance formula: \[ d = \sqrt{(5 - (-2))^2 + (-2 - 5)^2} \] 3. Simplify the terms inside the parentheses: \[ 5 - (-2) = 5 + 2 = 7 \] \[ -2 - 5 = -7 \] 4. Substitute these simplified values back into the equation: \[ d = \sqrt{(7)^2 + (-7)^2} \] \[ d = \sqrt{49 + 49} \] \[ d = \sqrt{98} \] 5. Simplify the radical value: \[ d = \sqrt{49 \times 2} = 7\sqrt{2} \]

Step 4: Final Answer:
The distance between the given coordinates is \(7\sqrt{2}\) units.
Therefore, the correct option is (A).