Question

A (1,1)
B (0,2)
C (2,4)
D (3,3)

Suppose you dilate the rectangle on the coordinate plane using a magnitude of 1/2 and the center of dilation at (1,1). What would be the corresponding ordered pair to the current point D?

The new ordered pair for point D' would be at (_,_)

Answer 1

To find the dilated point D' for point D (3, 3) using a center of dilation at (1, 1) and a magnitude of \( \frac{1}{2} \), we can use the following formula for dilation:

\[
D' = (1 + k \cdot (x - cx), 1 + k \cdot (y - cy))
\]

Where:
- \( (cx, cy) \) is the center of dilation (1, 1)
- \( (x, y) \) is the original point (3, 3)
- \( k \) is the dilation factor, which is \( \frac{1}{2} \)

Now we can substitute our values into the formula. The coordinates of point D are \( (x, y) = (3, 3) \), and the center of dilation is \( (cx, cy) = (1, 1) \).

Calculating the new x-coordinate:

\[
x' = 1 + \frac{1}{2} \cdot (3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2
\]

Calculating the new y-coordinate:

\[
y' = 1 + \frac{1}{2} \cdot (3 - 1) = 1 + \frac{1}{2} \cdot 2 = 1 + 1 = 2
\]

Therefore, the new ordered pair for point D' is:

\[
D' = (2, 2)
\]

So the answer would be:

The new ordered pair for point D' would be at \( (2, 2) \).