Question

1. higher order thinking the vertices of △abc are a(−5, 5), b(−2, 4), and c(−4, 2). △abc is reflected across the y - axis and then reflected again across the x - axis to produce the image △abc. what are the coordinates of △abc?

Explanation:

Step1: Recall reflection rules

• Reflection across the y - axis: For a point \((x,y)\), the image after reflection across the y - axis is \((-x,y)\).
• Reflection across the x - axis: For a point \((x,y)\), the image after reflection across the x - axis is \((x, -y)\).

Step2: Reflect point A(-5,5) across y - axis then x - axis

• Reflect across y - axis: Using the rule \((x,y)\to(-x,y)\), for \(A(-5,5)\), the image after y - axis reflection \(A_1=(5,5)\).
• Reflect \(A_1=(5,5)\) across x - axis: Using the rule \((x,y)\to(x, -y)\), the image \(A'=(5,-5)\).

Step3: Reflect point B(-2,4) across y - axis then x - axis

• Reflect across y - axis: For \(B(-2,4)\), using \((x,y)\to(-x,y)\), the image after y - axis reflection \(B_1=(2,4)\).
• Reflect \(B_1=(2,4)\) across x - axis: Using \((x,y)\to(x, -y)\), the image \(B'=(2,-4)\).

Step4: Reflect point C(-4,2) across y - axis then x - axis

• Reflect across y - axis: For \(C(-4,2)\), using \((x,y)\to(-x,y)\), the image after y - axis reflection \(C_1=(4,2)\).
• Reflect \(C_1=(4,2)\) across x - axis: Using \((x,y)\to(x, -y)\), the image \(C'=(4,-2)\).

Answer:

The coordinates of \(\triangle A'B'C'\) are \(A'(5, - 5)\), \(B'(2, - 4)\), \(C'(4, - 2)\)