https://file.aiquickdraw.com/mathbot/file/1725982158855bfjt89lz.jpg

The problem involves two transformations applied to the triangle \( \triangle JKL \):

1. **Dilation** by a scale factor of \( 0.5 \) centered at the origin.
2. **Translation** by 3 units **up**.

### Step 1: Original Coordinates of Triangle \( JKL \)
From the image, we can read the original coordinates of \( J \), \( K \), and \( L \):

- \( J (-6, 4) \)
- \( K (2, 8) \)
- \( L (0, -4) \)

### Step 2: Dilation
For a dilation with scale factor \( 0.5 \), multiply each coordinate by \( 0.5 \). The new coordinates \( J', K', L' \) after dilation will be:

- \( J' = (-6 \times 0.5, 4 \times 0.5) = (-3, 2) \)
- \( K' = (2 \times 0.5, 8 \times 0.5) = (1, 4) \)
- \( L' = (0 \times 0.5, -4 \times 0.5) = (0, -2) \)

### Step 3: Translation
Now, translate the dilated coordinates 3 units **up** by adding 3 to the y-coordinate of each point:

- \( J'' = (-3, 2 + 3) = (-3, 5) \)
- \( K'' = (1, 4 + 3) = (1, 7) \)
- \( L'' = (0, -2 + 3) = (0, 1) \)

### Final Answer:
The coordinates of the vertices of \( \triangle J''K''L'' \) are:
- \( J'' (-3, 5) \)
- \( K'' (1, 7) \)
- \( L'' (0, 1) \)

For the multiple-choice question asking for the coordinates of \( J'' \), the correct answer is \( (-3, 5) \).

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Would you like further details on any of the steps? Here are some follow-up questions you might consider:

1. What is the general formula for dilating a point?
2. How do you translate a shape along the y-axis or x-axis?
3. How does a dilation affect the area of a shape?
4. What would the coordinates of \( J'' \) be if the scale factor were \( 2 \)?
5. How does a negative scale factor affect a dilation?

**Tip:** When working with dilation, always remember the center of dilation,

Dario dilates triangle JKL in the coordinate plane using a scale factor of 0.5 and a center of dilation at the origin. He labels the image as triangle J'K'L'. Then he translates the image 3 units up to form triangle J''K''L''. What are the coordinates of the vertices of triangle J''K''L''?

Learn how to apply dilation with a scale factor of 0.5 and translation 3 units up to triangle JKL on the coordinate plane. This detailed step-by-step solution covers key transformation concepts such as dilation and translation. Suitable for students in Grades 7-9.

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