Question

The distance between points A (-5, 7) and B (-1, 3) is:

• A. 4 units
• B. 6 units
• C. \(4\sqrt{2}\) units
• D. 7 units

Hint:

When using the distance formula, be very careful with negative signs, especially when subtracting a negative coordinate. It's helpful to write out each step clearly to avoid errors, like \((-1 - (-5)) = (-1 + 5)\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the distance between two points in a Cartesian coordinate system, we use the distance formula, which is derived from the Pythagorean theorem.
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 formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Detailed Explanation:
Let the coordinates of point A be \( (x_1, y_1) = (-5, 7) \).
Let the coordinates of point B be \( (x_2, y_2) = (-1, 3) \).
Substitute these values into the distance formula:
\[ d = \sqrt{(-1 - (-5))^2 + (3 - 7)^2} \]
Be careful with the double negative sign:
\[ d = \sqrt{(-1 + 5)^2 + (-4)^2} \]
\[ d = \sqrt{(4)^2 + (-4)^2} \]
\[ d = \sqrt{16 + 16} \]
\[ d = \sqrt{32} \]
To simplify the square root of 32, find the largest perfect square factor of 32. \(32 = 16 \times 2\).
\[ d = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]
Step 4: Final Answer:
The distance between points A and B is \(4\sqrt{2}\) units.