Question

The distance between the points \(P(-4, 5)\) and \(Q(-1, 2)\) is

• A. 5
• B. \(3\sqrt{2}\)
• C. 6
• D. \(2\sqrt{3}\)

Hint:

Notice that the absolute difference in \(x\) coordinates is \(| -1 - (-4) | = 3\), and the absolute difference in \(y\) coordinates is \(| 2 - 5 | = 3\).
Whenever the absolute horizontal and vertical differences are equal (let's say they are \(k\)), the straight-line distance is always \(k\sqrt{2}\).
Since \(k = 3\) here, the distance is immediately \(3\sqrt{2}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to calculate the distance between two specified points in a 2D Cartesian plane.
The given coordinates are \(P(x_1, y_1) = (-4, 5)\) and \(Q(x_2, y_2) = (-1, 2)\).

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:

• Identify the coordinates from the problem:
\(x_1 = -4\), \(y_1 = 5\)
\(x_2 = -1\), \(y_2 = 2\)

• Substitute these coordinate values into the distance formula:
\[ d = \sqrt{(-1 - (-4))^2 + (2 - 5)^2} \]

• Simplify the expression inside the first set of parentheses:
\[ -1 - (-4) = -1 + 4 = 3 \]

• Simplify the expression inside the second set of parentheses:
\[ 2 - 5 = -3 \]

• Substitute these simplified values back into our square root equation:
\[ d = \sqrt{(3)^2 + (-3)^2} \]

• Compute the squares of the numbers:
\[ (3)^2 = 9 \]
\[ (-3)^2 = 9 \]

• Add the squared values:
\[ d = \sqrt{9 + 9} = \sqrt{18} \]

• Simplify the radical \(\sqrt{18}\) by factoring out the perfect square:
\[ d = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]


Step 4: Final Answer:
The distance between the two points is \(3\sqrt{2}\). This corresponds to option (B).