https://file.aiquickdraw.com/mathbot/file/17271025626634r2uparf.png

The task in the image is asking to reflect triangle \( \Delta TUV \) over the line \( y = x \). To solve this:

1. **Understand Reflection over the Line \( y = x \):**
   Reflecting a point \((x, y)\) across the line \(y = x\) switches the coordinates of the point to \((y, x)\).

2. **Coordinates of the Vertices:**
   The coordinates of \( T \), \( U \), and \( V \) from the image seem to be:
   - \( T \) looks like \( (-5, -6) \)
   - \( U \) looks like \( (-3, -6) \)
   - \( V \) looks like \( (-6, -8) \)

3. **Reflect the Points:**
   - Reflect \( T(-5, -6) \): Switch coordinates → \( T'(-6, -5) \)
   - Reflect \( U(-3, -6) \): Switch coordinates → \( U'(-6, -3) \)
   - Reflect \( V(-6, -8) \): Switch coordinates → \( V'(-8, -6) \)

4. **Plot the Reflected Triangle:**
   The new triangle \( T'U'V' \) will have the vertices:
   - \( T'(-6, -5) \)
   - \( U'(-6, -3) \)
   - \( V'(-8, -6) \)

By plotting these points on the graph, you can draw the reflected triangle.

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Would you like more details or further clarification on this reflection process? Here are five related questions:

1. How do you find the reflection of a point across the \(x\)-axis or \(y\)-axis?
2. What is the general formula for reflecting a point over any arbitrary line, not just \( y = x \)?
3. How does reflecting across \( y = -x \) differ from reflecting across \( y = x \)?
4. How does this transformation change the orientation of the triangle?
5. What are the practical applications of reflecting shapes in geometry?

**Tip:** When reflecting over \( y = x \), always switch the coordinates \( (x, y) \to (y, x) \).

Graph the image of ∆TUV after a reflection over the line y = x.

Learn how to reflect triangle ∆TUV over the line y = x using coordinate transformations. This step-by-step guide explains how to apply the formula (x, y) → (y, x) to transform points and graph the reflected image. Suitable for Grades 8-10.

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