The distance formula is derived from the Pythagorean theorem. When we have two points in a coordinate plane, we can form a right triangle where the line segment between the points is the hypotenuse. The horizontal and vertical distances become the legs of this triangle. Using the Pythagorean theorem a squared plus b squared equals c squared, we get the distance formula: d equals the square root of (x2 minus x1) squared plus (y2 minus y1) squared.
Now let's apply the distance formula to find the distance between the specific points (-4, 3) and (2, -1). We'll label these points A and B respectively. Point A has coordinates (-4, 3) and point B has coordinates (2, -1). Our goal is to calculate the straight-line distance between these two points using the distance formula.
Now we substitute the coordinates into the distance formula. For point A(-4, 3), x1 is -4 and y1 is 3. For point B(2, -1), x2 is 2 and y2 is -1. Substituting these values, we get d equals the square root of (2 minus negative 4) squared plus (negative 1 minus 3) squared. Simplifying the differences inside the parentheses, we have 2 plus 4 which is 6, and negative 1 minus 3 which is negative 4. So our formula becomes d equals the square root of 6 squared plus negative 4 squared.
Now we calculate the squares: 6 squared is 36, and negative 4 squared is 16. Adding these together gives us 52. So we have d equals the square root of 52. To simplify this, we look for perfect square factors of 52. Since 52 equals 4 times 13, and 4 is a perfect square, we can write the square root of 52 as the square root of 4 times 13, which equals the square root of 4 times the square root of 13. This simplifies to 2 times the square root of 13. As a decimal approximation, this is approximately 7.21.
Let's verify our result geometrically using the Pythagorean theorem. We've formed a right triangle with our two points, where the horizontal leg has length 6 and the vertical leg has length 4. According to the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the other two sides. So we have 6 squared plus 4 squared, which is 36 plus 16, equaling 52. Taking the square root of 52 gives us our distance of 2 times the square root of 13, confirming our previous calculation.
Let's solve this problem using an alternative approach - applying the Pythagorean theorem directly. We can visualize the problem by drawing a right triangle where the line segment between our two points is the hypotenuse. The horizontal distance between the points is 6 units, and the vertical distance is 4 units. These form the two legs of our right triangle. Using the Pythagorean theorem, a squared plus b squared equals c squared, we substitute our values: 6 squared plus 4 squared equals c squared. This gives us 36 plus 16 equals c squared, or 52 equals c squared. Taking the square root of both sides, we find that c equals the square root of 52, which simplifies to 2 times the square root of 13, matching our previous result.
Let's practice with another example. We need to find the distance between points (1, 5) and (-3, 2). Using the distance formula, we substitute our coordinates: d equals the square root of (negative 3 minus 1) squared plus (2 minus 5) squared. This simplifies to the square root of (negative 4) squared plus (negative 3) squared. Calculating the squares, we get the square root of 16 plus 9, which is the square root of 25. The square root of 25 is 5. So the distance between these two points is exactly 5 units. We can verify this by noting that we have a right triangle with legs of length 3 and 4, which forms a 3-4-5 Pythagorean triple, confirming that the hypotenuse has length 5.