https://file.aiquickdraw.com/mathbot/file/1727707659892fwc7yfgi.jpg

The problem you're working on involves two transformations applied to a triangle \( \Delta JKL \) in the coordinate plane:

1. **Dilation** with a **scale factor of 0.5** and a center of dilation at the origin.
2. **Translation** of the dilated image **3 units up** to get the final image \( \Delta J'K'L' \).

We need to determine the new coordinates of point \( J' \), particularly after these transformations.

Let's walk through this step by step:

### Original Coordinates:
From the diagram, the original coordinates of the triangle are:
- \( J(-6, 0) \)
- \( K(2, 8) \)
- \( L(4, -4) \)

### Step 1: Dilation by a factor of 0.5
To perform a dilation about the origin with a scale factor of 0.5, multiply each coordinate by 0.5.

For \( J \), which is \( (-6, 0) \):
- New \( x \)-coordinate: \( -6 \times 0.5 = -3 \)
- New \( y \)-coordinate: \( 0 \times 0.5 = 0 \)

So, the coordinates of \( J' \) after dilation are \( (-3, 0) \).

### Step 2: Translation 3 units up
To translate a point 3 units up, add 3 to the \( y \)-coordinate.

For \( J'(-3, 0) \), we add 3 to the \( y \)-coordinate:
- New \( y \)-coordinate: \( 0 + 3 = 3 \)

Thus, the final coordinates of \( J' \) are \( (-3, 3) \).

### Conclusion:
The coordinates of \( J' \) after both transformations are \( (-3, 3) \).

#### Multiple Choice Answer:
The correct coordinates of \( J' \) are not listed among the multiple-choice options provided, as none show \( (-3, 3) \).

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Would you like more details on any step, or have any additional questions?

Here are 5 relative questions to further expand this information:
1. How would the transformation differ if the scale factor was 2 instead of 0.5?
2. What would happen if the translation was 3 units down instead of up?
3. How do we apply a dilation with a center that is not the origin?
4. What is the general rule for translating points in any direction?
5. How can we determine the coordinates after a reflection instead of a translation?

**Tip**: When performing dilations, always remember to multiply both the \( x \)- and \( y \)-coordinates by the same scale factor.

Dario dilates triangle JKL in the coordinate plane using a scale factor of 0.5 and translates the image 3 units up. What are the coordinates of J'?

Learn how to find the coordinates of J' after dilating triangle JKL with a scale factor of 0.5 and translating it 3 units up. Understand the use of dilation and translation formulas in coordinate geometry.

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