To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on the coordinate plane using the Pythagorean theorem, we can treat the points as vertices of a right triangle, where the difference in $x$-coordinates and the difference in $y$-coordinates form the two legs of the triangle.

The Pythagorean theorem states:

$a^2 + b^2 = c^2$

Where:

• $a$ is the horizontal distance between the two points, which is $|x_2 - x_1|$,
• $b$ is the vertical distance between the two points, which is $|y_2 - y_1|$,
• $c$ is the distance between the two points, which is the hypotenuse of the right triangle.

Thus, the distance $d$ between the two points is given by:

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

Example:

If the points are $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$, the distance between them is:

$d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

So, the distance between the two points is 5 units.

Let me know if you'd like any more clarification, or if you want to work through another example!

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Here are some follow-up questions for you to explore:

1. How would you apply the distance formula to find the distance between two points with negative coordinates?
2. How can you derive the distance formula starting from the Pythagorean theorem?
3. How does the distance formula relate to the concept of vectors in geometry?
4. How does the Pythagorean theorem generalize to 3D space for distance calculation?
5. Can the distance formula be used in non-Euclidean geometries?

Tip: If you ever need to check your math when using the distance formula, try calculating the squares of differences separately first to avoid any mistakes.