What is the image point of left parenthesis, 6, comma, 6, right parenthesis(6,6) after the transformation r, start subscript, y, equals, minus, x, end subscript, circle, D, start subscript, one third, end subscriptr 
y=−x
​
 ∘D 
3
1
​
 
​
 ?

Let's break down the transformation step by step.

You are dealing with two transformations:
1. **Dilation by a factor of \( \frac{1}{3} \)**
   - Denoted as \( D_{\frac{1}{3}} \), this transformation shrinks the point by a factor of \( \frac{1}{3} \).
2. **Reflection over the line \( y = -x \)**
   - Denoted as \( r_{y = -x} \), this reflects a point across the line \( y = -x \).

### Step 1: Apply the dilation \( D_{\frac{1}{3}} \)
The dilation \( D_{\frac{1}{3}} \) scales both coordinates by \( \frac{1}{3} \).

Given point \( (6, 6) \), apply the dilation:
\[
D_{\frac{1}{3}}(6, 6) = \left( \frac{1}{3} \cdot 6, \frac{1}{3} \cdot 6 \right) = (2, 2)
\]

### Step 2: Apply the reflection \( r_{y = -x} \)
After dilation, we reflect the point \( (2, 2) \) across the line \( y = -x \). The reflection rule across \( y = -x \) swaps and negates the coordinates:
\[
r_{y = -x}(2, 2) = (-2, -2)
\]

### Final Answer:
The image point of \( (6, 6) \) after the transformation \( r_{y = -x} \circ D_{\frac{1}{3}} \) is \( (-2, -2) \).

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do you apply a dilation transformation with a different factor?
2. What is the general formula for reflecting points over a line?
3. How do composite transformations work geometrically?
4. What would happen if the dilation factor was negative?
5. How would you describe the composition of two reflections?

**Tip:** In a reflection over \( y = -x \), the coordinates of any point swap and change signs!

What is the image point of (6,6) after the transformation r_{y=-x} ∘ D_{1/3}?

Discover how to find the image point of (6,6) after a transformation involving a dilation by a factor of 1/3 followed by a reflection over the line y = -x. This problem explores geometric transformations, including dilation and reflection, and provides a step-by-step solution suitable for students in Grades 9-10.

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