Select all the sequences of reflections that produce an image equivalent to the image r(180°, O)(△BCD).

Answer:

r(180°,0) is a rotation of 180° degrees over the origin.

Notice that this rotation moves our figure to the opposite quadrant (so a translation of two quadrants).

Then this is equivalent to:

A reflection over the x-axis followed by a reflection over the y-axis.

Or.

A reflection over the y-axis followed by a reflection over the x-axis.

There is another possible reflection, but it depends on where is our figure.

If the figure is in the first or third quadrant, a reflection over the line y = -x is equivalent to the rotation.

If the figure is in the second or third quadrant, then the reflection over the line y = x is equivalent to the rotation.

We can combine those two and write:

A reflection over the line y = (-1)^n*x.

Where n is the number associated with the quadrant where the figure is in.

Final answer:

A rotation reflection, r(180°, O)(△BCD), can be achieved by performing two reflections over intersecting lines.

If the lines intersect at an angle of 90 degrees, the combination of the two reflections would result in a 180-degree rotation.

Explanation:

The mathematical question requires knowledge of geometrical transformations, specifically, reflections.

The rotation reflection, r(180°, O)(△BCD), means the initial triangle within the plane is reflected over a point 'O' by 180 degrees.

This reflection will result in an image equivalent to a series of two reflections over intersecting lines.

In commonly accepted mathematical conventions, it is generally accepted that any rotation can be represented by two reflections over intersecting lines.

For instance, two reflections over lines intersecting at an angle ∅/2 represent a rotation by an angle ∅, hence, a rotation by 180 degrees would mean the intersecting angle is 90 degrees.

Learn more about Geometrical Transformations here:

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