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Triangle XYZ is drawn on a coordinate plane with vertices X(0,4) , Y(2,0) , and Z(4,7) . If you dilate the triangle to form triangle X′Y′Z′  using a magnitude of 12 and a center of dilation at (4,7) , what are the coordinate points of the new vertices?(1 point)ResponsesX′(−4,−3), Y′(−2,−7), Z′(0,0)upper X prime left parenthesis negative 4 comma negative 3 right parenthesis ,  upper Y prime left parenthesis negative 2 comma negative 7 right parenthesis ,  upper Z prime left parenthesis 0 comma 0 right parenthesisX′(4,11), Y′(6,7), Z′(8,14)upper X prime left parenthesis negative 4 comma 11 right parenthesis ,  upper Y prime left parenthesis 6 comma 7 right parenthesis ,  upper Z prime left parenthesis 8 comma 14 right parenthesisX′(−4,1), Y′(0,−7), Z′(4,7)upper X prime left parenthesis negative 4 comma 1 right parenthesis ,  upper Y prime left parenthesis 0 comma negative 7 right parenthesis ,  upper Z prime left parenthesis 4 comma 7 right parenthesisX′(2,5.5), Y′(3,3.5), Z′(4,7)

1 answer

To find the coordinates of the new vertices \( X' \), \( Y' \), and \( Z' \) after dilating triangle \( XYZ \) using a center of dilation at point \( (4, 7) \) and a magnitude of 12, we can apply the dilation formula.

The formula for dilation from a center point \( (h, k) \) with a scale factor \( s \) for any point \( (x, y) \) is:

\[ (x', y') = (h + s \cdot (x - h), k + s \cdot (y - k)) \]

  1. Dilating Point \( X(0, 4) \):

    • Center: \( (h, k) = (4, 7) \)
    • Scale factor: \( s = 12 \) \[ X' = \left( 4 + 12 \cdot (0 - 4), 7 + 12 \cdot (4 - 7) \right) \] \[ X' = \left( 4 - 48, 7 - 36 \right) = (-44, -29) \]

  2. Dilating Point \( Y(2, 0) \): \[ Y' = \left( 4 + 12 \cdot (2 - 4), 7 + 12 \cdot (0 - 7) \right) \] \[ Y' = \left( 4 - 24, 7 - 84 \right) = (-20, -77) \]

  3. Dilating Point \( Z(4, 7) \):

    • Dilation of the center point does not change its coordinates since \( (x, y) = (h, k) \). \[ Z' = (4, 7) \]

  4. Final Coordinates: After computing the results based on the dilation formula:

    • \( X'(-44, -29) \)
    • \( Y'(-20, -77) \)
    • \( Z'(4, 7) \)

Based on these calculations, we can assert that none of the provided answers are correct. Please verify the dilation ratio and center point if needed, as I've applied the dilation formula correctly.