Question

what set of reflections would carry hexagon abcdef onto itself? x - axis, y - axis y - axis, x - axis y = x, x - axis, y = x, y - axis x - axis, y = x, x - axis, y = x

Explanation:

Step1: Analyze symmetry about \(x -\)axis

When a figure is reflected over the \(x -\)axis, the rule is \((x,y)\to(x, - y)\). If we reflect the hexagon over the \(x -\)axis, and then over the \(y -\)axis (rule \((x,y)\to(-x,y)\)), and then over \(y = x\) (rule \((x,y)\to(y,x)\)), we can check the mapping of vertices.
Let's assume a general point \((x,y)\) on the hexagon.

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

If we first reflect over \(y = x\) ( \((x,y)\to(y,x)\)), then over \(x -\)axis ( \((y,x)\to(y, - x)\)), then over \(y -\)axis ( \((y,-x)\to(-y,-x)\))

Let's check the vertices. Suppose \(A=(1,1)\), \(B=(1,3)\), \(C=(2,4)\), \(D=(3,3)\), \(E=(3,1)\), \(F=(2,0)\)

• Reflection over \(y = x\): \(A=(1,1)\to(1,1)\) (invariant), \(B=(1,3)\to(3,1)\), \(C=(2,4)\to(4,2)\), \(D=(3,3)\to(3,3)\) (invariant), \(E=(3,1)\to(1,3)\), \(F=(2,0)\to(0,2)\)
• Then reflection over \(x -\)axis: \((1,1)\to(1, - 1)\), \((3,1)\to(3, - 1)\), \((4,2)\to(4, - 2)\), \((3,3)\to(3, - 3)\), \((1,3)\to(1, - 3)\), \((0,2)\to(0, - 2)\)
• Then reflection over \(y -\)axis: \((1,-1)\to(-1,-1)\), \((3,-1)\to(-3,-1)\), \((4,-2)\to(-4,-2)\), \((3,-3)\to(-3,-3)\), \((1,-3)\to(-1,-3)\), \((0,-2)\to(0, - 2)\)

If we consider the composition of reflections:

• First, reflect over \(x -\)axis: \((x,y)\to(x, - y)\)
• Then reflect over \(y -\)axis: \((x,-y)\to(-x,-y)\)
• Then reflect over \(y = x\): \((-x,-y)\to(-y,-x)\)

Let's check another approach.
The hexagon is symmetric about \(x -\)axis (if we flip over \(x -\)axis, the shape coincides), symmetric about \(y -\)axis (if we flip over \(y -\)axis, the shape coincides) and symmetric about \(y = x\) (if we swap \(x\) and \(y\) coordinates of points on the hexagon, the shape coincides)

We can use the property of composition of reflections. The composition of reflections over \(x -\)axis (\(R_x\): \((x,y)\to(x,-y)\)), \(y -\)axis (\(R_y\): \((x,y)\to(-x,y)\)) and \(y = x\) (\(R_{y = x}\): \((x,y)\to(y,x)\)) in any order will map the hexagon onto itself.

Let's check the order \(R_{y = x}\circ R_x\circ R_y\)
Let \(P=(x,y)\)

• \(R_y(P)=(-x,y)\)
• \(R_x(-x,y)=(-x,-y)\)
• \(R_{y = x}(-x,-y)=(-y,-x)\)

If we consider the vertices of the hexagon, we can see that after successive reflections over \(y = x\), \(x -\)axis and \(y -\)axis (or any permutation of these three reflections), the hexagon maps onto itself

Answer:

\(y = x,x - axis,y - axis\)