Question

triangle abc is rotated 180° using the origin as the center of rotation.
which sequence of transformations will produce the same result?
a translation up 4 and then a reflection over the y - axis
a translation right 6 and then a reflection over the x - axis
a translation up 4 and then a translation right 6
a reflection over the x - axis and then a reflection over the y - axis

Explanation:

Step1: Find coordinates of original and rotated triangle

Original triangle \(ABC\) has coordinates \(A(-4, - 4)\), \(B(-2,-5)\), \(C(-2,-3)\). After \(180^{\circ}\) rotation about the origin, the rule \((x,y)\to(-x,-y)\) gives \(A'(4,4)\), \(B'(2,5)\), \(C'(2,3)\). But in the second figure (assuming it's the result of the transformation we are comparing), let's check the other option.
For the option “a reflection over the \(x -\)axis and then a reflection over the \(y -\)axis”:

• Reflection over the \(x -\)axis: The rule \((x,y)\to(x, - y)\). For \(A(-4,-4)\), \(A_1(-4,4)\); for \(B(-2,-5)\), \(B_1(-2,5)\); for \(C(-2,-3)\), \(C_1(-2,3)\).
• Reflection over the \(y -\)axis: The rule \((x,y)\to(-x,y)\). For \(A_1(-4,4)\), \(A_2(4,4)\); for \(B_1(-2,5)\), \(B_2(2,5)\); for \(C_1(-2,3)\), \(C_2(2,3)\).

Step2: Check other options

• For “a translation up \(4\) and then a reflection over the \(y -\)axis”:
• Translation up \(4\): \((x,y)\to(x,y + 4)\). \(A(-4,-4)\to A_1(-4,0)\), \(B(-2,-5)\to B_1(-2,-1)\), \(C(-2,-3)\to C_1(-2,1)\).
• Reflection over \(y -\)axis: \((x,y)\to(-x,y)\). \(A_1(-4,0)\to(4,0)

eq A'\) (from rotation).

• For “a translation right \(6\) and then a reflection over the \(x -\)axis”:
• Translation right \(6\): \((x,y)\to(x + 6,y)\). \(A(-4,-4)\to A_1(2,-4)\), \(B(-2,-5)\to B_1(4,-5)\), \(C(-2,-3)\to C_1(4,-3)\).
• Reflection over \(x -\)axis: \((x,y)\to(x,-y)\). \(A_1(2,-4)\to(2,4)

eq A'\) (from rotation).

• For “a translation up \(4\) and then a translation right \(6\)”:
• Translation up \(4\): \((x,y)\to(x,y + 4)\). \(A(-4,-4)\to(-4,0)\), \(B(-2,-5)\to(-2,-1)\), \(C(-2,-3)\to(-2,1)\).
• Translation right \(6\): \((x,y)\to(x + 6,y)\). \(A(-4,0)\to(2,0)

eq A'\) (from rotation).

Answer:

a reflection over the \(x -\)axis and then a reflection over the \(y -\)axis