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On a coordinate plane, a square and a point are shown. The square has points R prime (negative 8, 1), S prime (negative 4, 1), T prime (negative 4, negative 3), and U prime (negative 8, negative 3). Point S is at (3, negative 5).

Square RSTU is translated to form R'S'T'U', which has vertices R'(–8, 1), S'(–4, 1), T'(–4, –3), and U'(–8, –3). If point S has coordinates of (3, –5), which point lies on a side of the pre-image, square RSTU?

(–5, –3)
(3, –3)
(–1, –6)
(4, –9)

1 answer

To find which point lies on a side of the pre-image square \( RSTU \) with vertices \( R(-8, 1) \), \( S(-4, 1) \), \( T(-4, -3) \), and \( U(-8, -3) \), we first need to identify the coordinates of the sides of the square.

The sides of the square are as follows:

  1. From \( R(-8, 1) \) to \( S(-4, 1) \): This side is horizontal at \( y = 1 \) between \( x = -8 \) and \( x = -4 \).

  2. From \( S(-4, 1) \) to \( T(-4, -3) \): This side is vertical at \( x = -4 \) between \( y = 1 \) and \( y = -3 \).

  3. From \( T(-4, -3) \) to \( U(-8, -3) \): This side is horizontal at \( y = -3 \) between \( x = -4 \) and \( x = -8 \).

  4. From \( U(-8, -3) \) to \( R(-8, 1) \): This side is vertical at \( x = -8 \) between \( y = -3 \) and \( y = 1 \).

Now, we need to check the possible coordinates given in the options to see which point lies on one of these sides:

  1. Point: \( (-5, -3) \)

    • This point is not on the side since \( x = -5 \) is not in the range of \( x = -8 \) and \( x = -4 \) for \( y = -3 \).

  2. Point: \( (3, -3) \)

    • This point is also not on the side since \( x = 3 \) is not in the range of \( x = -8 \) and \( x = -4 \) for \( y = -3 \).

  3. Point: \( (-1, -6) \)

    • This point is not on the side since \( y = -6 \) is not in the range of \( y = -3 \) and \( y = 1 \).

  4. Point: \( (4, -9) \)

    • This point is not on the side since \( x = 4 \) is also not in the range of \( x = -8 \) and \( x = -4 \).

None of the points listed are on the sides of the square \( RSTU \). Is there any mistake in the provided options, or would more information be helpful?