Find the distance between the two points in simplest radical form.

(7,1) and (4,5)

To find the distance between the two points \((7, 1)\) and \((4, 5)\), we use the distance formula, which is derived from the Pythagorean theorem:

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

Here, \((x_1, y_1) = (7, 1)\) and \((x_2, y_2) = (4, 5)\).

Substituting these values into the formula:

\[
d = \sqrt{(4 - 7)^2 + (5 - 1)^2}
\]

Simplify the expressions inside the square root:

\[
d = \sqrt{(-3)^2 + (4)^2}
\]

\[
d = \sqrt{9 + 16}
\]

\[
d = \sqrt{25}
\]

\[
d = 5
\]

Thus, the distance between the points \((7, 1)\) and \((4, 5)\) is \(5\) units.

Would you like further details or have any questions? Here are some related questions:

1. How do you find the midpoint between the points \((7, 1)\) and \((4, 5)\)?
2. What is the distance between the points \((7, 1)\) and \((7, 5)\)?
3. How would the distance change if one point were moved along the x-axis?
4. Can the distance formula be used in three dimensions? How?
5. How is the distance formula related to the Pythagorean theorem?

**Tip:** When working with the distance formula, it's essential to ensure that you're squaring the differences correctly, as this will always result in a non-negative value under the square root.

Learn how to find the distance between two points (7, 1) and (4, 5) using the distance formula step-by-step. This problem covers coordinate geometry and the application of the Pythagorean theorem. Suitable for students in Grades 7-9.

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