To find which point lies on a side of the pre-image square RSTU, we first need to determine the position and dimensions of square RSTU based on the given translated square R'S'T'U'.
The vertices of square R'S'T'U' are:
- R'(–8, 1)
- S'(–4, 1)
- T'(–4, –3)
- U'(–8, –3)
From this, we can find the center of the square R'S'T'U' as the average of the coordinates of its vertices. The width of the square can be determined by the distance between S' and T'.
The length of the sides of the square R'S'T'U' can be calculated as follows:
- The distance between S' and T': \[ \text{Distance} = |y_{T'} - y_{S'}| = |-3 - 1| = 4 \]
Thus, the side length of the square is 4, indicating that square RSTU must have the same side lengths.
Next, we can find the original square RSTU by applying the inverse translation that transforms RSTU to R'S'T'U'.
To find the translation, we can observe the changes in the coordinates from RSTU to R'S'T'U'. If we denote the original coordinates of S as \(S(3, -5)\) and consider how \(S\) translates to \(S'\) at \((-4, 1)\), we can create the translation vector:
- From S to S':
- Change in x-coordinate: \( -4 - 3 = -7\)
- Change in y-coordinate: \( 1 - (-5) = 6\)
Thus, the translation vector is \((-7, 6)\).
Now we can find the original coordinates of square RSTU by applying the translation in the opposite direction.
Using point S(3, -5):
- For \(R\), \(S\), and \(T\) we reverse the translation:
- \[ R = (x, y) + (7, -6) \text{... (find from the pre-image neighbors)} \] Here, for point R where \(S'\) is \((-4, 1)\), we calculate it from \(S'(−4, 1)\) and however R maps out.
Without loss of generality, if we take the path: \[ S: (3, -5) \to S': (-4, 1) \] Thus, translating \(S\) back to the pre-image forms as: \[ (x + 7, y - 6) \]
For points lying on the original square, we check them against the translations previously observed. Side points would range amongst corners.
Calculating options against y = -3 as edges, then examining potential placements near S:
- \((-5, -3)\): Would fit a horizontal segment near \((-8, -3)~(x)\)
- \((3, -3)\): Would not intersect given linear layout.
- \((-1, -6)\): Would not fit edge structure.
- \((4, -9)\): Too far from square bounds.
Since the translated space suggests linear fits, only:
- Thus \((-5, -3)\) as the intersection hit formed by current sides.
Verifying against square definitions confirms validity. Thus: \[ \text{The point that lies on a side of the pre-image square RSTU is } (-5, -3). \]