Question

1. what is the result of two rotations of 180 degrees around the same point?

reflected figure
same as initial figure
translated figure
scaled figure

Explanation:

Step1: Recall rotation properties

A rotation of \(180^{\circ}\) around a point is a transformation. If we rotate a figure \(180^{\circ}\) around a point and then rotate the resulting figure \(180^{\circ}\) around the same point.
Let the rotation transformation be \(R_{O,180^{\circ}}\) (rotation about point \(O\) by \(180^{\circ}\)). If we have a point \(P(x,y)\) in a coordinate - plane, after the first rotation \(R_{O,180^{\circ}}(P)=P_1(-x,-y)\) (assuming \(O\) is the origin for simplicity of explanation, the general property holds for any center of rotation). After the second rotation \(R_{O,180^{\circ}}(P_1)=P(x,y)\)

In general, for a rotation \(R\) of \(180^{\circ}\) about a point \(O\), \(R\circ R\) (composition of the rotation with itself) is the identity transformation.

A reflection changes the orientation (left - right or top - bottom in a non - symmetric figure). A translation moves the figure without rotating or reflecting. A scaled figure changes the size of the figure.

Answer:

Same as initial figure