What is the distance between the points $(4,7)$ and $(4,-5)$ ?
To find the distance between two points on a Cartesian plane, we can analyze their coordinates. The two given points are \(A(4, 7)\) and \(B(4, -5)\). Since both points have the same x-coordinate, which is \(x = 4\), the line segment connecting them is a vertical line. The distance \(d\) between these two points is determined by the absolute difference of their y-coordinates.
The y-coordinate of point \(A\) is \(y_1 = 7\) and the y-coordinate of point \(B\) is \(y_2 = -5\). The distance formula for points with the same x-coordinate is:
\[d = |y_1 - y_2|\]
Substituting the known values into this formula gives:
\[d = |7 - (-5)|\]
Evaluating the expression inside the absolute value:
\[d = |7 + 5|\]
\[d = |12|\]
\[d = 12\]
Thus, the distance between the two points is 12 units. The correct option is the one corresponding to 12 units. The final answer is \(\boxed{12}\).
Finding the distance between points is a fundamental concept in coordinate geometry, essential for various applications such as calculating the length of a boundary in a plot plan or determining the distance between two landmarks on a map represented by a grid system. For instance, if you are planning a route between two points on a coordinate grid, understanding how to calculate the distance between them ensures accurate measurement of the path length, which is crucial for navigation and logistical planning.