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Suppose parallelogram ABCD is dilated using a magnitude of 2 and a center of dilation at (−2,−1) . What would be the corresponding ordered pair at point A ? (1 point) Responses (−4,3) left parenthesis negative 4 comma negative 3 right parenthesis (−1,2) left parenthesis negative 1 comma 2 right parenthesis (−5,0) left parenthesis negative 5 comma 0 right parenthesis (−3,0)

A's original position was (-3, 1)

1 answer

To find the new coordinates of point A after dilation, we follow these steps:

  1. Identify the center of dilation and the original point's coordinates.

    • Center of dilation: \( C(-2, -1) \)
    • Original coordinates of point A: \( A(-3, 1) \)

  2. Use the dilation formula. The formula for dilation is: \[ A' = C + k \cdot (A - C) \] where \( A' \) is the new point, \( C \) is the center of dilation, \( k \) is the dilation factor, and \( A \) is the original point.

  3. Calculate the vector from the center to point A: \[ A - C = (-3, 1) - (-2, -1) = (-3 + 2, 1 + 1) = (-1, 2) \]

  4. Multiply the vector by the dilation factor (which is 2): \[ k \cdot (A - C) = 2 \cdot (-1, 2) = (-2, 4) \]

  5. Add this value to the center of dilation to find the new point \( A' \): \[ A' = C + k \cdot (A - C) = (-2, -1) + (-2, 4) = (-2 - 2, -1 + 4) = (-4, 3) \]

Thus, the corresponding ordered pair at point A after dilation is \((-4, 3)\).

The correct response is \((-4, 3)\).