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A (-3,1) B(-5,-1) D (1,0) C(-1,-2)

Suppose parallelogram ABCD is dilated using a magnitude of 2 and a center of dilation at (-2,-1). What could be the corresponding ordered pair at point A

1 answer

To find the corresponding point \( A' \) after a dilation of point \( A (-3, 1) \) with a center of dilation at \( (-2, -1) \) and a magnitude of 2, we can use the formula for dilation:

\[ A' = (1 - k) O + k A \]

Where:

  • \( O \) is the center of dilation
  • \( k \) is the scale factor (in this case, \( k = 2 \))
  • \( A \) is the original point

Given:

  • \( O = (-2, -1) \)
  • \( A = (-3, 1) \)
  • \( k = 2 \)

Now we can calculate the coordinates of \( A' \):

  1. Calculate \( A - O \): \[ A - O = (-3, 1) - (-2, -1) = (-3 + 2, 1 + 1) = (-1, 2) \]

  2. Scale this vector by \( k = 2 \): \[ k(A - O) = 2 \cdot (-1, 2) = (-2, 4) \]

  3. Add this to the center of dilation \( O \): \[ A' = O + k(A - O) = (-2, -1) + (-2, 4) = (-2 - 2, -1 + 4) = (-4, 3) \]

Thus, the corresponding ordered pair for point \( A \) after the dilation is:

\[ \boxed{(-4, 3)} \]