The distance formula calculates the straight-line distance between two points in a coordinate plane. It's derived from the Pythagorean theorem, which relates the sides of a right triangle.
Let's find the distance between the points (-3, 4) and (2, -1). We'll identify these as our two points and plot them on a coordinate plane.
We identify the coordinates of our points. For Point 1 (-3, 4), x1 equals -3 and y1 equals 4. For Point 2 (2, -1), x2 equals 2 and y2 equals -1.
To visualize the distance calculation, we construct a right triangle. We draw a horizontal line from Point 1 and a vertical line from Point 2, creating the two legs of a right triangle. The hypotenuse is the line connecting our original points.
Now we calculate the differences in coordinates. The horizontal distance is x2 minus x1, which is 2 minus negative 3, equaling 5. The vertical distance is y2 minus y1, which is negative 1 minus 4, equaling negative 5. Since distance is always positive, we take the absolute value, giving us 5.
We substitute our calculated differences into the distance formula. D equals the square root of (5) squared plus (negative 5) squared. This simplifies to the square root of 25 plus 25, which equals the square root of 50.
Finally, we simplify the square root of 50. We factor 50 as 25 times 2. The square root of 25 times 2 equals the square root of 25 times the square root of 2. This simplifies to 5 times the square root of 2, which is approximately 7.07 units.
In summary, the distance between the points (-3, 4) and (2, -1) is 5 times the square root of 2, or approximately 7.07 units. We achieved this result by using the distance formula, which is derived from the Pythagorean theorem.