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: Identify the coordinates.
We have point \(P(-2, 5)\) and point \(Q(5, -2)\). So \(x_1 = -2,\, y_1 = 5,\, x_2 = 5,\, y_2 = -2\).
Step 2: Recall the distance formula.
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Compute the differences.
\(x_2 - x_1 = 5 - (-2) = 7\) and \(y_2 - y_1 = -2 - 5 = -7\).
Step 4: Square the differences.
\((7)^2 = 49\) and \((-7)^2 = 49\).
Step 5: Add and simplify.
\(d = \sqrt{49 + 49} = \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\).
Step 6: Select the correct option.
The distance is \(7\sqrt{2}\), which is option 1.
\[ \boxed{7\sqrt{2}} \]