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 ? (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)

Answer 1

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 \((-4, 3)\).