What is the distance between the points $(2,-5)$ and $(9,-5)$?
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a coordinate plane is given by the distance formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Given the points \((x_1, y_1) = (2, -5)\) and \((x_2, y_2) = (9, -5)\), we can substitute these values into the distance formula.
Since the y-coordinates are the same, \(y_1 = y_2 = -5\), the second term under the square root becomes zero:
\[d = \sqrt{(9 - 2)^2 + (-5 - (-5))^2}\]
\[d = \sqrt{(7)^2 + (0)^2}\]
\[d = \sqrt{49}\]
\[d = 7\]
Thus, the distance between the two points is \(7\) units.
Calculating the distance between points is fundamental in mapping and coordinate geometry. For example, if you are looking at a map on a grid and you want to know the distance between two locations that lie on the same latitude line (horizontal line), you simply find the difference in their longitude coordinates. This is exactly how we determined that the distance between the points \((2, -5)\) and \((9, -5)\) is \(7\) units.