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The slope of a line is a measure of its steepness and direction. It tells us how much the line rises or falls as we move from left to right on a graph. To find the slope between two points, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here,

• \(m\) represents the slope,
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points on the line.

For the points \((3, -1)\) and \((5, -7)\), you substitute into the formula: \[ m = \frac{-7 - (-1)}{5 - 3} = \frac{-6}{2} = -3 \] The slope is \(-3\). This means for every unit we move to the right, the line descends 3 units down. A negative slope indicates a downward trend from left to right.

The distance between two points on a coordinate plane is found using the distance formula. This formula is based on Pythagorean theorem and helps us calculate the straight-line distance between two points. The formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Where,

• \(d\) is the distance between the points,
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Using our points \((3, -1)\) and \((5, -7)\): \[ d = \sqrt{(5 - 3)^2 + (-7 - (-1))^2} = \sqrt{2^2 + (-6)^2} \] \[ d = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \] This calculation tells us that the length of the line segment linking these points is \(2\sqrt{10}\). This value represents the shortest distance between them on the plane.

A line segment is the part of a line that connects two points. Unlike a line, it does not extend infinitely in both directions. Instead, it has a definite length and two endpoints that bound it. When you are asked to find the length of a line segment, you're essentially measuring the shortest path between its endpoints on the coordinate plane. For example, in the problem, we are dealing with the line segment \(\overline{AB}\), connecting point \(A(3, -1)\) and point \(B(5, -7)\). The process involves analyzing properties like slope and distance, which help in understanding the segment's orientation and length. The slope tells us how steep the segment is and the distance formula calculates how long it is. Line segments are fundamental in geometry as they form the basic building blocks for more complex shapes like triangles and polygons. They help to define sides, create angles, and determine overall structure.