Question

select all sequences of translations, rotations, and reflections below that would take polygon
p to polygon q.
a. rotate ( 180^{circ} ) around point a.
b. translate so that a is taken to j. then reflect over line ba.
c. rotate ( 60^{circ} ) counterclockwise around point a and then reflect over the line fa.
d. reflect over the line ba and then rotate ( 60^{circ} ) counterclockwise around point a.
e. reflect over line ba and then translate by directed line segment ba.

1. here is triangle pog. match the description of the rotation

with the image of pog under that rotation.
a. rotate 60 degrees clockwise around o. c. rotate 60 degrees counterclockwise around o.
b. rotate 120 degrees clockwise around o. d. rotate 60 degrees clockwise around p.

Explanation:

Step1: Analyze rotation around \(O\)

• For a rotation of \(60^{\circ}\) clockwise around \(O\):
• The angle between the original and rotated figure around \(O\) for a \(60^{\circ}\) clockwise rotation.
• Let's consider the position of the vertices. If we rotate \(\triangle POG\) \(60^{\circ}\) clockwise around \(O\), we can observe the new position of the vertices.
• For a rotation of \(120^{\circ}\) clockwise around \(O\):
• The angle of rotation is larger. By visual - inspection of the grid and the center of rotation \(O\), we can determine the new vertex positions.
• For a rotation of \(60^{\circ}\) counter - clockwise around \(O\):
• Opposite direction of the \(60^{\circ}\) clockwise rotation around \(O\). Using the properties of rotation (preserving distances from the center of rotation \(O\) and the angle of rotation), we can analyze the vertex positions.

Step2: Analyze rotation around \(P\)

• For a rotation of \(60^{\circ}\) clockwise around \(P\):
• The center of rotation is \(P\). We use the rule that for a rotation of \(\theta\) around a point \(C(x_0,y_0)\), a point \(A(x,y)\) is transformed to \(A'(x_0+(x - x_0)\cos\theta+(y - y_0)\sin\theta,y_0-(x - x_0)\sin\theta+(y - y_0)\cos\theta)\). In the grid - based context, we can also use the fact that the distance from \(P\) to other vertices is preserved and the angle of \(60^{\circ}\) is considered.

Answer:

A. Rotate \(60\) degrees clockwise around \(O\) - \(4\)
B. Rotate \(120\) degrees clockwise around \(O\) - \(1\)
C. Rotate \(60\) degrees counterclockwise around \(O\) - \(3\)
D. Rotate \(60\) degrees clockwise around \(P\) - \(2\)