Question

what is a reflection rule that maps each triangle and its image?
the reflection rule is ( r_t(x,y)=square ), where the equation of line ( t ) is ( square ).
(simplify your answers.)

Explanation:

Step1: Recall the formula for reflection over the line \(y = -x\)

The formula for reflection over the line \(y=-x\) is \(r_{y = -x}(x,y)=(-y,-x)\).

Step2: Verify the line of reflection

To find the equation of the line of reflection, we note that for a point \((x,y)\) and its image \((-y,-x)\), the line of reflection is \(y=-x\). We can check this by taking a point, say \((1,2)\). Its reflection over \(y = -x\) is \((-2,-1)\). The mid - point of the segment joining \((1,2)\) and \((-2,-1)\) is \((\frac{1-2}{2},\frac{2 - 1}{2})=(-\frac{1}{2},\frac{1}{2})\), and the slope of the line joining \((1,2)\) and \((-2,-1)\) is \(\frac{2+1}{1 + 2}=1\), while the slope of \(y=-x\) is \(-1\). The product of the slopes of the line joining a point and its image and the line of reflection is \(- 1\) (since \(1\times(-1)=-1\)), which is a property of reflection.

Answer:

The reflection rule is \(r_{t}(x,y)=(-y,-x)\), where the equation of line \(t\) is \(y=-x\).