https://file.aiquickdraw.com/mathbot/file/1725909904353zdeu7rcy.jpg

To solve this problem, we need to apply a dilation transformation to the rectangle QRST with a scale factor of 4 and a center of dilation at the origin (0, 0).

### Given Coordinates:
- \( Q(-8, 5) \)
- \( R(-8, 7) \)
- \( S(-5, 7) \)
- \( T(-5, 5) \)

### Rule for Dilation:
In general, when you dilate a point \((x, y)\) by a scale factor of \(k\) with the center of dilation at the origin, the new coordinates \((x', y')\) are calculated as:
\[
(x', y') = (kx, ky)
\]
For this problem, the scale factor \(k = 4\). Thus, the transformation rule is:
\[
(x, y) \rightarrow (4x, 4y)
\]

### Applying the Dilation to Each Point:
Now, let's apply this transformation to each vertex of the rectangle QRST:

1. **For point \( Q(-8, 5) \):**
   \[
   Q' = (4 \times -8, 4 \times 5) = (-32, 20)
   \]
   
2. **For point \( R(-8, 7) \):**
   \[
   R' = (4 \times -8, 4 \times 7) = (-32, 28)
   \]
   
3. **For point \( S(-5, 7) \):**
   \[
   S' = (4 \times -5, 4 \times 7) = (-20, 28)
   \]
   
4. **For point \( T(-5, 5) \):**
   \[
   T' = (4 \times -5, 4 \times 5) = (-20, 20)
   \]

### Transformation Rule:
The rule for the transformation is:
\[
(x, y) \rightarrow (4x, 4y)
\]

Would you like further clarification on dilation or on how the transformation rule is applied?

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Here are 5 related questions to expand your understanding:
1. What happens if the center of dilation is not at the origin?
2. How do you reverse a dilation transformation?
3. How would a scale factor less than 1 affect the shape?
4. How do you calculate the area of the dilated rectangle compared to the original?
5. Can dilation change the orientation of a shape?

**Tip:** Always multiply both the x and y coordinates by the scale factor when dilating from the origin.

Rectangle QRST has coordinates Q(-8, 5), R(-8, 7), S(-5, 7), and T(-5, 5). Dilate the rectangle by a scale factor of 4 with a center of dilation at the origin. Write the rule to describe this transformation.

Learn how to apply a dilation transformation to a rectangle with coordinates Q(-8, 5), R(-8, 7), S(-5, 7), and T(-5, 5). Dilate the rectangle by a scale factor of 4 from the origin using the formula (x', y') = (kx, ky). Suitable for Grades 7-10.

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