Calculate the exact distance between the points (-2, 3) and (4, 9).

A.)

12
B.)

2 square root of 10
C.)

5 square root of 2
D.)

6 square root of 2

To calculate the exact distance between the points \((-2, 3)\) and \( (4, 9) \), we use the distance formula:

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

Plugging in the coordinates \((-2, 3)\) and \((4, 9)\):

\[ x_1 = -2, \quad y_1 = 3 \]
\[ x_2 = 4, \quad y_2 = 9 \]

Calculate the differences:

\[ x_2 - x_1 = 4 - (-2) = 4 + 2 = 6 \]
\[ y_2 - y_1 = 9 - 3 = 6 \]

Substitute these into the distance formula:

\[ d = \sqrt{(6)^2 + (6)^2} \]
\[ d = \sqrt{36 + 36} \]
\[ d = \sqrt{72} \]
\[ d = \sqrt{36 \times 2} \]
\[ d = \sqrt{36} \times \sqrt{2} \]
\[ d = 6 \sqrt{2} \]

So, the exact distance between the points is \(6 \sqrt{2}\).

Therefore, the correct answer is:

D.) \(6 \sqrt{2}\)

Would you like more details or have any questions? Here are some additional questions to expand your understanding:

1. What is the distance formula used to calculate the distance between two points in a plane?
2. How do you derive the distance formula from the Pythagorean theorem?
3. Can the distance formula be applied in three-dimensional space? If so, how?
4. What are the coordinates of the midpoint between \((-2, 3)\) and \( (4, 9)\)?
5. How would you calculate the slope of the line segment connecting \((-2, 3)\) and \( (4, 9)\)?
6. What is the equation of the line passing through the points \((-2, 3)\) and \( (4, 9)\)?
7. How do you determine if two points are collinear with a third point?
8. How does the distance formula change if the points are represented in polar coordinates?

**Tip:** Always simplify your answers to their most reduced form for clarity and precision.

Learn how to calculate the exact distance between the points (-2, 3) and (4, 9) using the distance formula. This problem involves understanding coordinate geometry, applying the distance formula, and simplifying square roots. Suitable for Grades 9-12 students.

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