Question

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

Answer 1

To find the corresponding ordered pair for point A after the dilation with a center at (-2, -1) and a magnitude of 2, we can use the formula for dilation. The formula for a point \( P(x, y) \) after dilation about a center point \( C(x_c, y_c) \) with a scale factor \( k \) is given by:

\[
P' = (x_c + k(x - x_c), y_c + k(y - y_c))
\]

Given:
- Point A: \( (-3, 1) \)
- Center of dilation: \( (-2, -1) \)
- Scale factor: \( k = 2 \)

Now we can plug the coordinates into the formula.

1. Calculate the new x-coordinate:
\[
x' = -2 + 2(-3 - (-2)) = -2 + 2(-3 + 2) = -2 + 2(-1) = -2 - 2 = -4
\]

2. Calculate the new y-coordinate:
\[
y' = -1 + 2(1 - (-1)) = -1 + 2(1 + 1) = -1 + 2(2) = -1 + 4 = 3
\]

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

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