How to calculate the length of a segment on the Cartesian plane

The Cartesian , also known as the plane, is a fundamental concept in mathematics. It consists of two perpendicular number lines, called the x-axis and y-axis, which intersect at a point called the origin. The Cartesian plane allows us to identify the position of any point using a set of coordinates.

When it comes to measuring the length of a segment on the Cartesian plane, we can utilize a simple formula derived from the Pythagorean theorem. This widely applicable theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

To calculate the length of a segment on the Cartesian plane, we need to determine the coordinates of the two endpoints of the segment. Let’s say the coordinates of one endpoint are (x₁, y₁) and the coordinates of the other endpoint are (x₂, y₂).

The first step is to calculate the difference between the x-coordinates and the difference between the y-coordinates of the two endpoints. We can do this by subtracting the x-coordinate of one endpoint from the x-coordinate of the other endpoint, and the same for the y-coordinate.

Next, we square each of these differences. This is because the Pythagorean theorem requires us to sum the squares of the two sides, as mentioned earlier.

After squaring the differences, we add them together. This gives us the sum of the squares of the side lengths.

Finally, we take the square root of the sum of the squared side lengths to find the length of the segment. This is because we need the actual length and not the square of it.

In mathematical terms, we can use the following formula to find the length of a segment on the Cartesian plane:

Length = √((x₂ – x₁)² + (y₂ – y₁)²)

Let’s consider an example to illustrate this further. Suppose we have two points, A (2, 4) and B (5, 8). To calculate the length of the segment AB, we plug the coordinates into the formula:

Length = √((5 – 2)² + (8 – 4)²)
Length = √(3² + 4²)
Length = √(9 + 16)
Length = √25
Length = 5

Thus, the length of segment AB on the Cartesian plane is 5 units.

This formula can be applied to measure the length of any segment on the Cartesian plane, regardless of the orientation or position of the segment. It is efficient and enables us to accurately calculate distances between two points without needing to physically measure them.

In conclusion, calculating the length of a segment on the Cartesian plane involves using the Pythagorean theorem and a simple formula. By finding the differences between the x-coordinates and y-coordinates of the two endpoints, squaring these differences, summing the squared side lengths, and taking the square root of the sum, we can determine the length of any segment on the Cartesian plane. This calculation method is a valuable tool in various mathematical disciplines and is essential for accurately measuring distances in a two-dimensional coordinate .