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: Recall rotation and transformation rules

A \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\). Let's assume \(A(-3,4)\), \(B(-2,5)\), \(C(-2,3)\). After \(180^{\circ}\) rotation, \(A'(3,-4)\), \(B'(2,-5)\), \(C'(2,-3)\).

Step2: Analyze each transformation option

• Option 1: Translation up 4 and then reflection over \(y -\)axis

If we first translate \(A(-3,4)\) up 4: \(A_1(-3,8)\), then reflect over \(y -\)axis: \(A_2(3,8)
eq A'(3,-4)\)

• Option 2: Translation right 6 and then reflection over \(x -\)axis

First translate \(A(-3,4)\) right 6: \(A_1(3,4)\), then reflect over \(x -\)axis: \(A_2(3,-4)\). For \(B(-2,5)\): translate right 6 to \(B_1(4,5)\), reflect over \(x -\)axis: \(B_2(4,-5)
eq B'(2,-5)\)

• Option 3: Translation up 4 and then translation right 6

Translate \(A(-3,4)\) up 4: \(A_1(-3,8)\), then right 6: \(A_2(3,8)
eq A'(3,-4)\)

• Option 4: Reflection over \(x -\)axis and then reflection over \(y -\)axis

Reflect \(A(-3,4)\) over \(x -\)axis: \(A_1(-3,-4)\), then reflect over \(y -\)axis: \(A_2(3,-4)\). Reflect \(B(-2,5)\) over \(x -\)axis: \(B_1(-2,-5)\), then over \(y -\)axis: \(B_2(2,-5)\). Reflect \(C(-2,3)\) over \(x -\)axis: \(C_1(-2,-3)\), then over \(y -\)axis: \(C_2(2,-3)\)

Answer:

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