A pentagon with vertices at (1,1), (2,3), (4,3), (5,1), and (3,0) is transformed to a pentagon with vertices at (1,−1), (2,−3), (4,−3), (5,−1), and (3,0). Which transformation was applied?

Question

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1 point
Rotation 180 degrees about the origin
Rotation 90 degrees counterclockwise about the origin
Translation 2 units down
Reflection over the x-axis

Answer 1

To determine the transformation applied to the pentagon, let's analyze the change from the original vertices to the new vertices.

The original vertices are:

• \(A(1, 1)\)
• \(B(2, 3)\)
• \(C(4, 3)\)
• \(D(5, 1)\)
• \(E(3, 0)\)

The transformed vertices are:

• \(A'(1, -1)\)
• \(B'(2, -3)\)
• \(C'(4, -3)\)
• \(D'(5, -1)\)
• \(E'(3, 0)\)

Now, let's compare the y-coordinates of the original vertices with those of the transformed vertices:

• For \(A\): \(1\) transforms to \(-1\) (change of \( -2\))
• For \(B\): \(3\) transforms to \(-3\) (change of \( -6\))
• For \(C\): \(3\) transforms to \(-3\) (change of \( -6\))
• For \(D\): \(1\) transforms to \(-1\) (change of \( -2\))
• For \(E\): \(0\) stays \(0\) (no change)

We can see that the y-coordinates of the transformed vertices are the negatives of the corresponding original vertices except for \(E\), which remains unchanged. This indicates a reflection over the x-axis.

Therefore, the correct transformation applied to the pentagon is:

Reflection over the x-axis.