Questions: In the xy-plane, points R, S, and T have coordinates (5,6),(5,1), and (8,1), respectively. How much greater is the distance from R to S than the distance from S to T?

To solve this problem, we need to calculate the distances between the given points in the \(xy\)-plane and then find the difference between these distances.

1. Distance from \(R\) to \(S\):

   The coordinates of points \(R\) and \(S\) are \(R(5, 6)\) and \(S(5, 1)\). Since these points have the same \(x\)-coordinate, the distance between them is simply the difference in their \(y\)-coordinates.

   \[ \text{Distance from } R \text{ to } S = |6 - 1| = 5 \]

2. Distance from \(S\) to \(T\):

   The coordinates of points \(S\) and \(T\) are \(S(5, 1)\) and \(T(8, 1)\). Since these points have the same \(y\)-coordinate, the distance between them is simply the difference in their \(x\)-coordinates.

   \[ \text{Distance from } S \text{ to } T = |8 - 5| = 3 \]

3. Difference in distances:

   Now, we find how much greater the distance from \(R\) to \(S\) is compared to the distance from \(S\) to \(T\).

   \[ \text{Difference} = 5 - 3 = 2 \]

Therefore, the distance from \(R\) to \(S\) is 2 units greater than the distance from \(S\) to \(T\).