Question

which reflection will produce an image of δrst with a vertex at (2, −3)?
○ a reflection of δrst across the x-axis
○ a reflection of δrst across the y-axis
○ a reflection of δrst across the line y = x
○ a reflection of δrst across the line y = −x
(graph with vertices r(-1, 3), s(3, -2), t(1, -4) on a coordinate plane)

Explanation:

Step1: Recall reflection rules

• Reflection over \(x\)-axis: \((x,y)\to(x,-y)\)
• Reflection over \(y\)-axis: \((x,y)\to(-x,y)\)
• Reflection over \(y = x\): \((x,y)\to(y,x)\)
• Reflection over \(y=-x\): \((x,y)\to(-y,-x)\)

Step2: Test each reflection for vertices

• Reflection over \(x\)-axis:
• \(R(-1,3)\to(-1,-3)\), \(S(3,-2)\to(3,2)\), \(T(1,-4)\to(1,4)\). No vertex at \((2,-3)\).
• Reflection over \(y\)-axis:
• \(R(-1,3)\to(1,3)\), \(S(3,-2)\to(-3,-2)\), \(T(1,-4)\to(-1,-4)\). No vertex at \((2,-3)\).
• Reflection over \(y = x\):
• \(R(-1,3)\to(3,-1)\), \(S(3,-2)\to(-2,3)\), \(T(1,-4)\to(-4,1)\). No vertex at \((2,-3)\).
• Reflection over \(y=-x\):
• \(R(-1,3)\to(-3,1)\), \(S(3,-2)\to(2,-3)\), \(T(1,-4)\to(4,-1)\). Here, \(S(3,-2)\) reflects to \((2,-3)\).

Answer:

a reflection of \(\triangle RST\) across the line \(y = -x\)