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!

Learn how to calculate the distance between two points (-5, 7) and (7, 0) using the distance formula. This example demonstrates the application of the Pythagorean theorem in coordinate geometry. Suitable for students in Grades 8-10.

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