How do you calculate the length of a line segment?

To calculate the length of a line segment, use the distance formula derived from Pythagoras' theorem.

The distance formula is used to find the length of a line segment between two points in a coordinate plane. If you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the formula is:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula comes from Pythagoras' theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the context of a coordinate plane, the difference in the x-coordinates \((x_2 - x_1)\) and the difference in the y-coordinates \((y_2 - y_1)\) form the two shorter sides of a right-angled triangle. The distance between the points is the hypotenuse.

For example, if you have points \((3, 4)\) and \((7, 1)\), you can substitute these into the formula:

\[ \text{Distance} = \sqrt{(7 - 3)^2 + (1 - 4)^2} \]
\[ \text{Distance} = \sqrt{4^2 + (-3)^2} \]
\[ \text{Distance} = \sqrt{16 + 9} \]
\[ \text{Distance} = \sqrt{25} \]
\[ \text{Distance} = 5 \]

So, the length of the line segment between the points \((3, 4)\) and \((7, 1)\) is 5 units. This method can be used for any pair of points in a coordinate plane.