What is the length of the line segment with endpoints (10.8, -5.3) and (-15.6, -5.3)? Enter your answer as a decimal.
The problem asks for the length of a line segment connecting two points in a Cartesian coordinate system, \(P_1 = (10.8, -5.3)\) and \(P_2 = (-15.6, -5.3)\). To find the length, we observe that both points share the same y-coordinate (\(-5.3\)), which indicates that the line segment is horizontal. In such a case, the distance between the two points simplifies to the absolute difference of their x-coordinates.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the general distance formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Since the y-coordinates are equal, \(y_1 = y_2 = -5.3\), the term \((y_2 - y_1)^2\) becomes \(0\). The formula reduces to:
\[d = \sqrt{(x_2 - x_1)^2} = |x_2 - x_1|\]
Substituting the x-coordinates \(x_1 = 10.8\) and \(x_2 = -15.6\) into the simplified formula, we get:
\[d = |(-15.6) - 10.8| = |-26.4|\]
Taking the absolute value, the length of the segment is \(26.4\).
Calculating the distance between two points on a horizontal line is very useful in architecture and engineering when measuring the distance between two components that are mounted at the same height on a wall, or determining the distance between two trees planted along a straight, flat path. Understanding this simple distance calculation helps in precise layout and planning in various real-world projects. For example, if you are hanging two pictures at the same height of 5.3 feet from the floor, and their horizontal positions are 10.8 feet and -15.6 feet from a reference point, the distance between them is \(\boxed{26.4 \text{ feet}}\).