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Part 1

Find the distance between the pair of points.

Upper E left parenthesis 9 comma negative 3 right parenthesisE(9,−3)​, 

Upper F left parenthesis negative 3 comma 6 right parenthesis

To find the distance between the points \( E(9, -3) \) and \( F(-3, 6) \), we'll use the distance formula:

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

Given:
- \( x_1 = 9 \), \( y_1 = -3 \)
- \( x_2 = -3 \), \( y_2 = 6 \)

Substituting these values into the formula:

\[
d = \sqrt{(-3 - 9)^2 + (6 - (-3))^2}
\]

Simplifying the expression:

\[
d = \sqrt{(-12)^2 + (6 + 3)^2}
\]
\[
d = \sqrt{144 + 9^2}
\]
\[
d = \sqrt{144 + 81}
\]
\[
d = \sqrt{225}
\]
\[
d = 15
\]

So, the distance between the points \( E(9, -3) \) and \( F(-3, 6) \) is **15 units**.

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How do you find the slope of a line between two points?
2. What is the equation of a line passing through two given points?
3. How can the distance formula be derived from the Pythagorean theorem?
4. What are collinear points, and how do you determine if three points are collinear?
5. How do you calculate the area of a triangle using its vertices?

**Tip:** Remember that the distance formula is derived from the Pythagorean theorem, making it a powerful tool for measuring straight-line distances in a coordinate plane.

Learn how to find the distance between two points, E(9, -3) and F(-3, 6), using the distance formula. This problem involves basic coordinate geometry concepts and calculations. Suitable for students in Grades 7-9.

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