https://file.aiquickdraw.com/mathbot/file/1726773208841m13tf7ev.jpg

Let's go over the details from the image of the triangle transformation problem.

### Given:
1. A right triangle \( TUV \) is placed on a coordinate plane.
2. The triangle is dilated by a factor of \( \frac{1}{2} \), centered at the origin.
3. After dilation, it is 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. First, calculate the area of the original triangle \( TUV \).
   The area of a triangle is given by:
   \[
   \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   From the image, it looks like the base and height can be determined from the coordinates of points \( T \), \( U \), and \( V \). 

2. After dilation by \( \frac{1}{2} \), the new triangle’s dimensions are scaled down. Since the area scales by the square of the dilation factor, the area of \( T'U'V' \) will be:
   \[
   \text{New Area} = \left(\frac{1}{2}\right)^2 \times \text{Original Area}
   \]

### Part B:
**Find the length of the new side \( U'V' \).**

To determine this, first calculate the length of side \( UV \) using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Then apply the dilation factor to find \( U'V' \).

---

Let me go ahead and extract the necessary coordinates and solve these.### Solution:

1. **Part A:**
   - The area of the original triangle \( TUV \) is \( 16 \) square units.
   - After dilation by a factor of \( \frac{1}{2} \), the area of the new triangle \( T'U'V' \) is:
     \[
     \boxed{4} \text{ square units}.
     \]

2. **Part B:**
   - The length of side \( UV \) is approximately \( 5.66 \) units.
   - After dilation by \( \frac{1}{2} \), the length of the new side \( U'V' \) is:
     \[
     \boxed{2.83} \text{ units}.
     \]

Do you want further details on any of these parts or clarifications?

---

### Related questions:
1. How do transformations like translation affect the area of geometric shapes?
2. What other transformations can be applied to a triangle besides dilation and translation?
3. How is the distance formula derived, and how does it apply to different geometric shapes?
4. Can you derive the relationship between dilation factors and area changes in general?
5. How does the centroid of a triangle behave under dilation and translation?

### Tip:
Always remember that when a shape is dilated by a factor \( k \), the area scales by \( k^2 \), while the perimeter scales by \( k \).

Right triangle TUV, shown in the coordinate plane below, 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'? What is the length of the new side U'V'?

This problem involves transforming triangle TUV using dilation by a factor of 1/2 and translation to find the area of the new triangle and the length of side U'V'. Key concepts include coordinate geometry, transformation theorems, and the distance formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation