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To solve this problem, we need to reflect each of the vertices of the quadrilateral over the line \( y = x \). When reflecting a point \((x, y)\) over the line \( y = x \), the coordinates of the point change by swapping \(x\) and \(y\). That is, the reflection of \((x, y)\) over \(y = x\) becomes \((y, x)\).

Let's first determine the coordinates of the original points \(U\), \(V\), \(S\), and \(T\), and then find their reflections.

1. **Coordinates of the original points:**
   - Point \(U\) appears to be at \((2, -2)\).
   - Point \(V\) appears to be at \((-4, -2)\).
   - Point \(S\) appears to be at \((-2, -6)\).
   - Point \(T\) appears to be at \((6, -4)\).

2. **Reflection over the line \(y = x\):**
   - Reflect \(U(2, -2)\) to get \((-2, 2)\).
   - Reflect \(V(-4, -2)\) to get \((-2, -4)\).
   - Reflect \(S(-2, -6)\) to get \((-6, -2)\).
   - Reflect \(T(6, -4)\) to get \((-4, 6)\).

So, the coordinates of the vertices after the reflection over the line \(y = x\) are:

- \(U' = (-2, 2)\)
- \(V' = (-2, -4)\)
- \(S' = (-6, -2)\)
- \(T' = (-4, 6)\)

Would you like further details on the reflection process? Here are some related questions:

1. How do you reflect a point over the \(x\)-axis or \(y\)-axis?
2. What happens to coordinates when reflected over the line \(y = -x\)?
3. How can reflections help in determining symmetry in geometric figures?
4. What are some real-life applications of geometric reflections?
5. How do other transformations, such as rotations or translations, compare to reflections?

**Tip:** Always check the line of reflection before applying transformations to ensure the correct mapping of points.

Write the coordinates of the vertices after a reflection over the line y = x.

Learn how to find the coordinates of a quadrilateral's vertices after reflecting them over the line y = x. This solution includes step-by-step transformations of each vertex and their reflected coordinates. Suitable for students in Grades 8-10.

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