https://file.aiquickdraw.com/mathbot/file/1729550210489frhs2mks.jpg

We are given the rectangular coordinates of \(ABCD\) and told that it undergoes a dilation with a scale factor of \(\frac{1}{2}\), followed by a translation of 3 units downward. We need to find the coordinates of the transformed point \( D'' \).

### Step 1: Determine the coordinates of \( D \)
From the graph, we can see the coordinates of point \( D \) are \( (-7, 1) \).

### Step 2: Dilation by a scale factor of \(\frac{1}{2}\)
The dilation is centered at the origin, so to find the dilated coordinates of \( D \) (denoted as \( D' \)), we multiply both coordinates of \( D \) by \(\frac{1}{2}\):
\[
D' = \left( \frac{-7}{2}, \frac{1}{2} \right) = \left( -3.5, 0.5 \right)
\]

### Step 3: Translation 3 units downward
To translate \( D' \) down by 3 units, we subtract 3 from the \( y \)-coordinate of \( D' \):
\[
D'' = \left( -3.5, 0.5 - 3 \right) = \left( -3.5, -2.5 \right)
\]

Thus, the coordinates of vertex \( D'' \) are:
\[
D'' (-3.5, -2.5)
\]

Would you like more details on any part of this solution? 

Here are 5 related questions for further exploration:
1. What are the coordinates of vertex \( C'' \) after the dilation and translation?
2. How would the transformation change if the center of dilation were not at the origin?
3. What would be the coordinates of point \( D' \) if the scale factor were \(\frac{2}{3}\) instead of \(\frac{1}{2}\)?
4. How would the result differ if the rectangle was translated 3 units upwards instead of downwards?
5. How do you determine the new area of the rectangle after dilation?

**Tip:** Always apply the dilation first before translation or other transformations to get the correct result.

Angel dilates the photograph, rectangle ABCD, by a scale factor of 1/2 with the center of dilation at the origin to form rectangle A'B'C'D', and then translates rectangle A'B'C'D' 3 units down to form rectangle A''B''C''D''. What are the coordinates of vertex D''?

Learn how to find the coordinates of vertex D'' after dilating a rectangle by a factor of 1/2 and translating it 3 units down on a coordinate plane. This problem covers key concepts in dilation, translation, and coordinate geometry, suitable for Grades 7-9 students.

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