Question

question 3 points 2 figure abcd is rotated clockwise about the origin through 360°, and then reflected across the x-axis. it is then reflected across the y-axis. identify the transformed figure.

Explanation:

Step1: Analyze rotation by \(360^\circ\)

A \(360^\circ\) rotation about the origin leaves the figure unchanged. So after rotation, \(ABCD\) remains as is.

Step2: Reflect across \(x\)-axis

For a point \((x,y)\), reflection over \(x\)-axis gives \((x, -y)\). Apply this to each vertex of \(ABCD\).

Step3: Reflect across \(y\)-axis

For a point \((x,y)\) (after \(x\)-axis reflection), reflection over \(y\)-axis gives \((-x, y)\) (wait, no: after \(x\)-axis reflection, the point is \((x, -y)\); then reflecting over \(y\)-axis: \((-x, -y)\)). So overall, reflecting over \(x\) then \(y\) is equivalent to reflecting over the origin (since \((x,y)\to(x,-y)\to(-x,-y)\), which is a \(180^\circ\) rotation or reflection over origin). But let's track vertices:

• Let's find coordinates (from graph):
• \(A\): \((1, 2)\) (approx? Wait, looking at grid: \(A\) is at \((1, 2)\)? Wait, no, the grid: \(x\) and \(y\) axes. Let's see: \(B\) is at \((3, 0)\)? Wait, no, the graph: \(B\) is on \(x\)-axis at \(x = 3\)? Wait, maybe better to see:
• \(A\): (1, 2) (since between \(x=0\) and \(x=2\), \(y=2\))
• \(B\): (3, 0) (on \(x\)-axis, \(x=3\))
• \(C\): (2, -2)
• \(D\): (-2, 1)

Wait, maybe I misread. Let's re-express:
After \(360^\circ\) rotation: no change.
Reflect over \(x\)-axis: \((x,y)\to(x,-y)\)

• \(A(1,2)\to(1,-2)\)
• \(B(3,0)\to(3,0)\) (since \(y=0\))
• \(C(2,-2)\to(2,2)\)
• \(D(-2,1)\to(-2,-1)\)

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

• \(A(1,-2)\to(-1,-2)\)
• \(B(3,0)\to(-3,0)\)
• \(C(2,2)\to(-2,2)\)
• \(D(-2,-1)\to(2,-1)\)

Wait, but maybe the key is that reflecting over \(x\) then \(y\) is equivalent to a central inversion (reflection over origin). Alternatively, the transformed figure should be the one obtained by these reflections. But since the problem is to identify the figure (assuming options, but since the original figure is given and we do transformations, the final figure is the one that is the reflection of the original over both axes (i.e., each vertex \((x,y)\) becomes \((-x,-y)\)). So the transformed figure is the one that is the mirror image over both axes, which is the figure symmetric to \(ABCD\) with respect to the origin.

Answer:

The transformed figure is the one obtained by reflecting \(ABCD\) across the \(x\)-axis then \(y\)-axis (or over the origin), which would be the figure with vertices \((-1,-2)\), \((-3,0)\), \((-2,2)\), \((2,-1)\) (approx), but since the problem is to identify it (assuming options, but since the user's graph is shown, the final figure is the one that is the reflection over both axes. However, since the exact figure isn't listed here (as the user's image is the original), but in typical problems, the transformed figure after \(360^\circ\) rotation, \(x\)-axis reflection, \(y\)-axis reflection is the figure that is the reflection of \(ABCD\) over the origin. So the answer is the figure congruent to \(ABCD\) but mirrored over both axes (i.e., each point \((x,y)\) becomes \((-x,-y)\)).