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 legs are the horizontal and vertical distances between the points. The distance between the points is the hypotenuse of this triangle. Using the Pythagorean theorem, we get the distance formula: d equals the square root of the sum of the squares of the differences in x-coordinates and y-coordinates.
Now let's apply this to our specific problem. We're given one point with coordinates (-5, y1), but we need both points to calculate the distance. To use the distance formula, we must identify the x and y coordinates of both points. Let's call the first point P1 with coordinates (x1, y1) where x1 is -5, and the second point P2 with coordinates (x2, y2). We need to know the complete coordinates of both points to proceed with our calculation.
Now we substitute our known coordinates into the distance formula. Since x1 is -5, we replace x1 with -5 in the formula. This gives us d equals the square root of (x2 minus negative 5) squared plus (y2 minus y1) squared. It's crucial to handle the double negative correctly. When we subtract a negative number, it's the same as adding the positive value. So x2 minus negative 5 becomes x2 plus 5. This step is often where mistakes occur, so we must be careful with our signs.
When simplifying the distance formula, we must follow the order of operations. First, we handle the expressions inside the parentheses. As we saw, subtracting a negative number is the same as adding the positive number. So x2 minus negative 5 becomes x2 plus 5. Next, we square each term. It's important to remember that squaring any real number always results in a non-negative value. This is why the distance formula will always give us a positive result, which makes sense since distance cannot be negative. Finally, we add the squared terms and take the square root to find the distance.
Let's work through a complete example. We'll find the distance between (-5, 3) and (2, 7). First, we identify our coordinates: x1 is -5, y1 is 3, x2 is 2, and y2 is 7. Substituting into the formula, we get d equals the square root of (2 minus negative 5) squared plus (7 minus 3) squared. Simplifying inside the parentheses, 2 minus negative 5 is 2 plus 5 which is 7, and 7 minus 3 is 4. So we have the square root of 7 squared plus 4 squared. Calculating the squares, 7 squared is 49 and 4 squared is 16. Adding these gives us 65. Therefore, the distance is the square root of 65, which is approximately 8.06 units.
To verify our calculation, we can apply the Pythagorean theorem directly to the right triangle formed by our two points. The horizontal leg has length 7 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 c squared equals 7 squared plus 4 squared, which is 49 plus 16, or 65. Taking the square root gives us c equals the square root of 65, confirming our distance formula result. The distance formula has numerous practical applications. It's used in navigation systems to calculate distances between locations, in computer graphics for rendering 3D scenes, in engineering for structural design, and in physics for calculating vector magnitudes. When working with distance calculations, remember to be careful with signs, especially when dealing with negative coordinates, and always verify that your answer makes geometric sense.