Question

The coordinates of rectangle ABCD are A(0,0), B(6,0), C(6,4), and D(0,4). If the rectangle were dilated using a magnitude of 3 and center of dilation at (3,0), what would be the corresponding ordered pair to point A?

Answer 1

To perform the dilation of point \( A(0,0) \) using a center of dilation at \( (3,0) \) with a magnitude of 3, we can use the dilation formula:

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

where:
- \((x, y)\) are the original coordinates of the point,
- \((x_c, y_c)\) are the coordinates of the center of dilation,
- \(k\) is the magnitude of the dilation,
- \((x', y')\) are the coordinates after dilation.

Given:
- Point \( A \) is \( (0, 0) \).
- Center of dilation \( C \) is \( (3, 0) \).
- Magnitude of dilation \( k = 3 \).

Now we can substitute in these values:

- \( x = 0 \), \( y = 0 \)
- \( x_c = 3 \), \( y_c = 0 \)

Plugging these into the formula gives us:

\[
(x', y') = \left( 3 + 3(0 - 3), 0 + 3(0 - 0) \right)
\]

Calculating \( x' \):

\[
x' = 3 + 3(0 - 3) = 3 + 3(-3) = 3 - 9 = -6
\]

Calculating \( y' \):

\[
y' = 0 + 3(0 - 0) = 0 + 3(0) = 0
\]

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

\[
(-6, 0)
\]

So the final result is:

\[
\boxed{(-6, 0)}
\]