Calculating the Length of a Line Segment using the Distance Formula

Understanding how to calculate the length of a line segment can be useful in various mathematical and real-world applications. This article provides a detailed explanation of how to do so using the distance formula, as well as an alternative method using the Pythagorean theorem.

Introduction to the Distance Formula

The distance formula is a fundamental tool in geometry that helps determine the length of a line segment between two points on a coordinate plane. Given two points, (x1, y1) and (x2, y2), the distance D between these points is calculated as:

D √[(x2 - x1)2 (y2 - y1)2]

Application of the Distance Formula

Let us apply this formula to find the length of the line segment joining the points -2, -3 and 1, 4.

Identify the coordinates of the two points: x1 -2, y1 -3 and x2 1, y2 4. Substitute these values into the formula:

D √[(1 - (-2))2 (4 - (-3))2]

Simplify the equation:

D √[(1 2)2 (4 3)2] D √[32 72]

Calculate the squares and add them:

D √[9 49] D √58

Therefore, the length of the line segment is √58 units, approximately 7.62 units when rounded.

Alternative Method: Using the Pythagorean Theorem

Another approach to finding the length of a line segment involves interpreting it as the hypotenuse of a right-angled triangle. The legs of this triangle can be determined from the coordinates of the points.

Identify the horizontal and vertical distances. From the points -2, -3 and 1, 4, the horizontal distance (base) is 3 units (1 - (-2)) and the vertical distance (height) is 7 units (4 - (-3)). Apply the Pythagorean theorem (a2 b2 c2), where a and b are the legs of the triangle, and c is the hypotenuse:

c2 32 72 c2 9 49 c2 58

Solve for c:

c √58 c 7.62 (approx)

Hence, the length of the line segment is √58 units, approximately 7.62 units.

Conclusion

Both the distance formula and the Pythagorean theorem provide accurate methods to calculate the length of a line segment. Understanding these concepts is crucial in various fields, including mathematics, physics, and engineering. It's always a good idea to verify your calculations and consider multiple approaches to ensure accuracy.