To find the distance between two points on a coordinate plane, you can use the distance formula. The distance formula is derived from the Pythagorean theorem and calculates the straight-line distance between two points.

Formula:

If the coordinates of the two points are $(x_1, y_1)$ and $(x_2, y_2)$, then the distance $d$ between the points is given by:

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

Steps:

1. Identify the coordinates: Find the coordinates of the two points. Label them as $(x_1, y_1)$ and $(x_2, y_2)$.
2. Substitute the values: Plug the values into the distance formula.
3. Simplify: Perform the subtraction, square the differences, add them together, and take the square root of the sum to find the distance.

Example:

Let’s say you have two points: $A(3, 4)$ and $B(7, 1)$.

1. $x_1 = 3, y_1 = 4$
2. $x_2 = 7, y_2 = 1$

Substitute these into the formula:

$d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

So, the distance between points $A(3, 4)$ and $B(7, 1)$ is 5 units.

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Would you like a more detailed explanation or have any further questions?

Here are some related questions you might explore:

1. What is the derivation of the distance formula?
2. How do you find the midpoint of two points?
3. How does the distance formula apply to 3-dimensional space?
4. What are the applications of the distance formula in geometry?
5. How can you use the distance formula in real-world problems, such as navigation or architecture?

Tip: If you're calculating distance on a graph, it's often helpful to first visualize the points and count the horizontal and vertical distances to ensure accuracy.