Calculating Distance Between Coordinate Points

The question requires finding the distance between two points given in a coordinate plane. Point A has coordinates $(-5, 7)$ and Point B has coordinates $(-1, 3)$. We can use the distance formula, a standard tool in coordinate geometry, to solve this.

Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem ($a^2 + b^2 = c^2$). It helps find the length of the hypotenuse (the distance, $d$) of a right triangle formed by the horizontal and vertical distances between two points $(x_1, y_1)$ and $(x_2, y_2)$. The formula is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Here's what the variables represent:

• $d$ is the distance we want to calculate.
• $(x_1, y_1)$ are the coordinates of the first point (Point A).
• $(x_2, y_2)$ are the coordinates of the second point (Point B).

Applying the Formula to Points A and B

Given points:

• Point A: $(x_1, y_1) = (-5, 7)$
• Point B: $(x_2, y_2) = (-1, 3)$

Let's substitute these coordinates into the distance formula step-by-step:

1. Find the difference in the x-coordinates: $ \Delta x = x_2 - x_1 = -1 - (-5) $ $ \Delta x = -1 + 5 = 4 $
2. Find the difference in the y-coordinates: $ \Delta y = y_2 - y_1 = 3 - 7 $ $ \Delta y = -4 $
3. Square both differences: $ (\Delta x)^2 = (4)^2 = 16 $ $ (\Delta y)^2 = (-4)^2 = 16 $
4. Sum the squared differences: $ (\Delta x)^2 + (\Delta y)^2 = 16 + 16 = 32 $
5. Take the square root of the sum: $ d = \sqrt{32} $

Simplifying the Result

The distance is $\sqrt{32}$. We can simplify this radical by finding the largest perfect square factor of 32, which is 16.

$ d = \sqrt{16 \times 2} $

Using the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$:

$ d = \sqrt{16} \times \sqrt{2} $

$ d = 4 \times \sqrt{2} = 4\sqrt{2} $

Therefore, the distance between points A and B is $4\sqrt{2}$ units.

Comparing Calculation with Options

The calculated distance is $4\sqrt{2}$ units. Let's examine the provided options:

The calculation matches Option 3.