Question

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

Answer 1

To find the coordinates of a point after dilation, you use the formula:

\[
P' = O + k(P - O)
\]

where:
- \( P \) is the original point,
- \( P' \) is the dilated point,
- \( O \) is the center of dilation,
- \( k \) is the dilation factor.

In this case, the center of dilation \( O \) is (1, 1), the dilation factor \( k \) is \( \frac{1}{2} \), and we will apply this to point \( D(3, 3) \).

### Step 1: Calculate the difference \( P - O \)

Subtract the coordinates of \( O \) from the coordinates of \( D \):

\[
D - O = (3, 3) - (1, 1) = (3 - 1, 3 - 1) = (2, 2)
\]

### Step 2: Multiply by the dilation factor \( k \)

Now, multiply the result by \( k = \frac{1}{2} \):

\[
k(D - O) = \frac{1}{2}(2, 2) = \left(\frac{1}{2} \cdot 2, \frac{1}{2} \cdot 2\right) = (1, 1)
\]

### Step 3: Add the center of dilation \( O \)

Finally, add the center of dilation \( O \) back to this result:

\[
P' = O + k(D - O) = (1, 1) + (1, 1) = (1 + 1, 1 + 1) = (2, 2)
\]

Therefore, the corresponding ordered pair for point \( D(3, 3) \) after dilation is:

\[
\boxed{(2, 2)}
\]