The problem provided asks for the reflection rule for a triangle with given coordinates. The triangle's vertices and their reflected counterparts are as follows:

• Triangle vertices: $A(3,5), B(5,6), C(7,3)$
• Reflected triangle vertices: $A'(-5, -3), B'(-6, -5), C'(-3, -7)$

Step 1: Identifying the line of reflection

The reflection of a point across a line results in the point and its reflection being equidistant from the line, along a perpendicular. To find the equation of the line, we observe the original and reflected points.

One effective method is to find the midpoints between corresponding points, since the midpoint will lie on the line of reflection.

Midpoints between original and reflected points:

1. Midpoint of $A(3, 5)$ and $A'(-5, -3)$: $M_A = \left( \frac{3 + (-5)}{2}, \frac{5 + (-3)}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1)$

2. Midpoint of $B(5, 6)$ and $B'(-6, -5)$: $M_B = \left( \frac{5 + (-6)}{2}, \frac{6 + (-5)}{2} \right) = \left( \frac{-1}{2}, \frac{1}{2} \right) = (-0.5, 0.5)$

3. Midpoint of $C(7, 3)$ and $C'(-3, -7)$: $M_C = \left( \frac{7 + (-3)}{2}, \frac{3 + (-7)}{2} \right) = \left( \frac{4}{2}, \frac{-4}{2} \right) = (2, -2)$

Step 2: Equation of the line

We now have three points $(-1, 1), (-0.5, 0.5), (2, -2)$ on the line of reflection. The slope of the line passing through any two of these points can be calculated:

Using points $(-1, 1)$ and $(2, -2)$: $\text{slope} = \frac{-2 - 1}{2 - (-1)} = \frac{-3}{3} = -1$

The line has a slope of $-1$, and it passes through the point $(-1, 1)$. Using the point-slope form of the equation of a line:

$y - 1 = -1(x - (-1)) = -1(x + 1)$

Simplifying: $y = -x$

Thus, the line of reflection is $y = -x$.

Reflection Rule

The reflection rule across the line $y = -x$ is: $r_{y = -x}(x, y) = (-y, -x)$

This rule applies to all points of the triangle, confirming the transformation from the original to the reflected points.

Would you like further clarification or have any questions? Here are some related questions you might find useful:

1. What happens to a triangle's orientation after reflection across a line?
2. How do you find the line of reflection between two corresponding points?
3. What is the general form of the reflection rule for any line?
4. Can reflections change the size of geometric figures?
5. How do you reflect a point across vertical or horizontal lines?

Tip: When reflecting a point across a line, always check the perpendicular distance between the original and reflected point from the line to verify accuracy.