To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a plane, we use the distance formula:

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

Given points:

• Point 1: $(1, 3)$
• Point 2: $(4, 7)$

Calculation:

1. Subtract the $x$-coordinates: $x_2 - x_1 = 4 - 1 = 3$
2. Subtract the $y$-coordinates: $y_2 - y_1 = 7 - 3 = 4$
3. Square the differences: $(x_2 - x_1)^2 = 3^2 = 9, \quad (y_2 - y_1)^2 = 4^2 = 16$
4. Add the squares: $9 + 16 = 25$
5. Take the square root: $d = \sqrt{25} = 5$

Final Answer:

The distance between the points $(1, 3)$ and $(4, 7)$ is $\mathbf{5}$.

Would you like further clarification or details? 😊

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Related Questions:

1. How do you derive the distance formula?
2. What is the distance between $(-2, 5)$ and $(3, -1)$?
3. Can this method be used for points in 3D space?
4. What is the midpoint of the points $(1, 3)$ and $(4, 7)$?
5. How does this formula relate to the Pythagorean theorem?

Tip: Always label points clearly in coordinate geometry problems to avoid confusion.