Question

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

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

Hint:

Notice that the absolute horizontal difference is \(| 5 - (-2) | = 7\), and the absolute vertical difference is \(| -2 - 5 | = 7\).
Whenever the horizontal difference and vertical difference are equal (let's say they are \(k\)), the straight-line distance is always \(k\sqrt{2}\).
Since \(k = 7\) in this case, the distance is immediately \(7\sqrt{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the straight-line distance between two specified points in a 2D Cartesian plane.
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 any 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:

• Write down the coordinate values from the problem:
\(x_1 = -2\), \(y_1 = 5\)
\(x_2 = 5\), \(y_2 = -2\)

• Substitute these values into the distance formula:
\[ d = \sqrt{(5 - (-2))^2 + (-2 - 5)^2} \]

• Simplify the terms inside the parentheses:
- For the first term:
\[ 5 - (-2) = 5 + 2 = 7 \] - For the second term:
\[ -2 - 5 = -7 \]

• Substitute these values back into our equation:
\[ d = \sqrt{(7)^2 + (-7)^2} \]

• Calculate the squares of the numbers:
\[ (7)^2 = 49 \] \[ (-7)^2 = 49 \]

• Sum the squared values inside the radical:
\[ d = \sqrt{49 + 49} = \sqrt{98} \]

• Simplify the radical expression by factoring out the perfect square:
\[ d = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2} \]


Step 4: Final Answer:
The distance between the two points is \(7\sqrt{2}\). This corresponds to option (A).