Question

what is the rule for the reflection?
o ry=x(x,y) → (-y,-x)
o ry=-x(x,y) → (-y,-x)
o ry=x(x,y) → (y,x)
o ry=-x(x,y) → (y,x)

Explanation:

Step1: Recall reflection rules

• Reflection over the line \(y = x\) has the rule \((x,y)\to(y,x)\).
• Reflection over the line \(y=-x\) has the rule \((x,y)\to(-y, -x)\).

Step2: Analyze the coordinates

• Let's assume a point \(A(-1,0)\) in the original triangle. Its image \(A'(0, - 1)\).
• If we use the rule \((x,y)\to(-y,-x)\) for \(x=-1,y = 0\), we get \((-0,-(-1))=(0,1)\) (wrong).
• If we use the rule \((x,y)\to(-y,-x)\) for reflection over \(y=-x\).
• Let's take another point \(B(-3,4)\). Using the rule \((x,y)\to(-y,-x)\), we get \((-4,3)\) (wrong).
• Let's use the rule \((x,y)\to(y,x)\) for reflection over \(y = x\). For \(A(-1,0)\), we get \((0,-1)\). For \(B(-3,4)\), we get \((4,-3)\). For \(C(-4,1)\), we get \((1,-4)\) (wrong).
• Let's use the rule \((x,y)\to(-y,-x)\) for reflection over \(y=-x\).
• Let's check the general property of reflection. The line \(y=-x\) is the line of reflection.
• The formula for reflection over the line \(y =-x\) is \((x,y)\to(-y,-x)\).

Answer:

\(r_{y =-x}(x,y)\to(-y,-x)\) (the second option)