What is the length of the segment between (1, 1) and (4, 5)?

The length of the segment between (1, 1) and (4, 5) is 5 units.

To find the length of the segment between two points on a coordinate plane, we use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by:

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

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. For our specific points, \((1, 1)\) and \((4, 5)\), we can substitute these values into the formula:

\[ x_1 = 1, \, y_1 = 1, \, x_2 = 4, \, y_2 = 5 \]

Now, calculate the differences:

\[ x_2 - x_1 = 4 - 1 = 3 \]
\[ y_2 - y_1 = 5 - 1 = 4 \]

Next, square these differences:

\[ (x_2 - x_1)^2 = 3^2 = 9 \]
\[ (y_2 - y_1)^2 = 4^2 = 16 \]

Add these squared differences together:

\[ 9 + 16 = 25 \]

Finally, take the square root of the sum to find the distance:

\[ \sqrt{25} = 5 \]

So, the length of the segment between the points (1, 1) and (4, 5) is 5 units. This method can be used for any two points on a coordinate plane to find the distance between them.