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Quadrilateral RUST has a vertex at T (5, 2).

An image displaying a quadrilateral RUST.The vertex of R is R of 1 comma 5.
What are the coordinates of T' after the translation (x, y) → (x – 1, y + 3), followed by a dilation by a scale factor of 2, centered at the origin?

A.
(8, 10)
B.
(10, 4)
C.
(4, 5)
D.
(6, 8)

1 answer

To find the coordinates of vertex T after the given transformations, we must first apply the translation, followed by the dilation.

  1. Translation: The translation rule is given as \( (x, y) \rightarrow (x - 1, y + 3) \).

    For point T (5, 2):

    • New x-coordinate: \( 5 - 1 = 4 \)
    • New y-coordinate: \( 2 + 3 = 5 \)

    So after translation, the coordinates of T are \( T' = (4, 5) \).

  2. Dilation: The dilation with a scale factor of 2, centered at the origin, is given by the transformation \( (x, y) \rightarrow (2x, 2y) \).

    Applying this dilation to the translated coordinates \( T' (4, 5) \):

    • New x-coordinate: \( 2 \times 4 = 8 \)
    • New y-coordinate: \( 2 \times 5 = 10 \)

Thus, the coordinates of T' after the translation and dilation are \( (8, 10) \).

The correct answer is A. (8, 10).