Question

The distance between the points \((2, -3)\) and \((-2, 3)\) is:

• A. \(2\sqrt{13}\) units
• B. \(5\) units
• C. \(13\sqrt{2}\) units
• D. \(10\) units

Answer 1

The Correct Option is A

Step 1: Understanding the problem:
We are given two points: \( (2, -3) \) and \( (-2, 3) \), and we need to find the distance between these two points.

Step 2: Using the distance formula:
The formula to calculate the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where: - \( (x_1, y_1) = (2, -3) \), - \( (x_2, y_2) = (-2, 3) \).

Step 3: Substituting the values into the distance formula:
Substitute \( x_1 = 2 \), \( y_1 = -3 \), \( x_2 = -2 \), and \( y_2 = 3 \) into the formula:
\[ d = \sqrt{((-2) - 2)^2 + (3 - (-3))^2} \] Simplify the terms inside the parentheses:
\[ d = \sqrt{(-4)^2 + (6)^2} \] \[ d = \sqrt{16 + 36} \] \[ d = \sqrt{52} \] Now, simplify \( \sqrt{52} \):
\[ d = \sqrt{4 \times 13} = 2\sqrt{13} \]

Step 4: Conclusion:
The distance between the points \( (2, -3) \) and \( (-2, 3) \) is \( 2\sqrt{13} \) units.