The distance formula is used to find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane. The formula is:

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

Given the points $(-5, 7)$ and $(7, 0)$:

1. $x_1 = -5$, $y_1 = 7$
2. $x_2 = 7$, $y_2 = 0$

Now, substitute these values into the formula:

$d = \sqrt{(7 - (-5))^2 + (0 - 7)^2}$

Simplify inside the parentheses:

$d = \sqrt{(7 + 5)^2 + (-7)^2}$

$d = \sqrt{12^2 + (-7)^2}$

$d = \sqrt{144 + 49}$

$d = \sqrt{193}$

Therefore, the distance between the points $(-5, 7)$ and $(7, 0)$ is:

$d = \sqrt{193} \approx 13.89$

Would you like more details or have any questions?

Here are some related questions:

1. What is the midpoint between two points $(-5, 7)$ and $(7, 0)$?
2. How can the distance formula be used in 3D space?
3. What is the equation of a line passing through the points $(-5, 7)$ and $(7, 0)$?
4. Can the distance formula be derived from the Pythagorean theorem?
5. What is the slope of the line between $(-5, 7)$ and $(7, 0)$?

Tip: Remember that the distance formula is essentially an application of the Pythagorean theorem!