Which expression gives the length of the line segment between (-13, 5) and (7, 5)?
The problem asks for an expression that represents the length of the line segment connecting the points \(P_1 = (-13, 5)\) and \(P_2 = (7, 5)\). In a Cartesian coordinate system, the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Here, we have \((x_1, y_1) = (-13, 5)\) and \((x_2, y_2) = (7, 5)\).
Since the y-coordinates of both points are identical (\(y_1 = y_2 = 5\)), the line segment is horizontal. For horizontal segments, the distance simplifies to the absolute difference between the x-coordinates:
\[d = |x_2 - x_1|\]
Substituting the coordinates \(x_1 = -13\) and \(x_2 = 7\), we get:
\[d = |7 - (-13)| = |7 + 13| = |20| = 20\]
Alternatively, calculating the absolute difference in the order \(x_1 - x_2\) gives:
\[d = |-13 - 7| = |-20| = 20\]
Both expressions are equivalent and represent the length of the segment.
We compare our derived expression \(|-13 - 7|\) with the given options:
The expression in option (b) matches our calculated distance expression \(|-13 - 7|\), which equals 20. Therefore, (b) is the correct expression. The final answer is \(\boxed{b}\).
Understanding the distance between points is fundamental in map reading and city planning. For instance, if you are navigating a city grid where streets are numbered along an x-axis, calculating the distance between two locations on the same street (a horizontal line) is simply the difference between their addresses. If a store is at address -13 and another is at address 7, knowing the distance \(|-13 - 7| = 20\) blocks helps you plan your walk efficiently.