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The right triangle \( TUV \) in the coordinate plane is dilated by a factor of \( \frac{1}{2} \) centered at the origin, and then translated 3 units to the right to form a new triangle \( T'U'V' \).

### Part A: Find the Area of the New Triangle \( T'U'V' \)

1. **Identify the coordinates of triangle \( TUV \):**
   From the graph:
   - \( T(0, 4) \)
   - \( U(4, 4) \)
   - \( V(0, 0) \)

2. **Calculate the area of triangle \( TUV \):**
   The area of a triangle is given by:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   For triangle \( TUV \):
   - The base is the horizontal distance between \( V \) and \( U \), which is 4 units.
   - The height is the vertical distance between \( V \) and \( T \), which is 4 units.

   Thus, the area of \( TUV \) is:
   \[
   \text{Area of } TUV = \frac{1}{2} \times 4 \times 4 = 8 \text{ square units.}
   \]

3. **Apply the dilation factor:**
   The dilation factor is \( \frac{1}{2} \). The area of the new triangle is affected by the square of the dilation factor.
   \[
   \text{Area of } T'U'V' = \text{Area of } TUV \times \left(\frac{1}{2}\right)^2 = 8 \times \frac{1}{4} = 2 \text{ square units.}
   \]

4. **Conclusion:**
   The area of the new triangle \( T'U'V' \) is **2 square units**.

Would you like to go over Part B or any specific details? 

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Here are some additional questions that build on this concept:

1. What are the coordinates of the vertices of the new triangle \( T'U'V' \) after dilation and translation?
2. How does a dilation centered at a point other than the origin affect the coordinates of the triangle?
3. What happens to the area of a triangle if the dilation factor is greater than 1?
4. Can you generalize the formula for how area changes under dilation?
5. How does translation affect the size and area of geometric figures?

**Tip:** When a figure is dilated, the area changes by the square of the dilation factor. Keep this in mind for scaling problems!

Right triangle TUV, shown in the coordinate plane, is dilated by a factor of 1/2, centered at the origin, and then translated 3 units to the right to form a new triangle T'U'V'. What is the area of the new triangle T'U'V'?

This problem explores the dilation and translation of a right triangle TUV on a coordinate plane. Learn how to calculate the area of the new triangle after it is dilated by a factor of 1/2 and translated 3 units to the right. Suitable for Grade 8 students.

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