Find the distance between the points (4,5) and (-2,-3) in simplest radical form.

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is given by the distance formula:

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

For the points \((4, 5)\) and \((-2, -3)\), we substitute these coordinates into the formula:

\[
d = \sqrt{(-2 - 4)^2 + (-3 - 5)^2}
\]
\[
d = \sqrt{(-6)^2 + (-8)^2}
\]
\[
d = \sqrt{36 + 64}
\]
\[
d = \sqrt{100}
\]
\[
d = 10
\]

Thus, the distance between the points \((4, 5)\) and \((-2, -3)\) is \(10\).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the distance formula change in 3D space?
2. What is the midpoint of the segment connecting \((4, 5)\) and \((-2, -3)\)?
3. Can you find the slope of the line passing through \((4, 5)\) and \((-2, -3)\)?
4. What is the equation of the line through \((4, 5)\) and \((-2, -3)\)?
5. How does the distance formula relate to the Pythagorean theorem?

**Tip:** Always check whether a given distance can be simplified into a radical form by factoring out perfect squares.

Learn how to calculate the distance between two points, (4, 5) and (-2, -3), using the distance formula. This example illustrates how the Pythagorean theorem applies to coordinate geometry and provides a step-by-step solution. Suitable for students in Grades 6-8.

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