The hexagon GIKMPR is regular. The dashed line segments form 30° angles.
Hexagon GIKMPR is shown.A hexagon is shown. A point in the middle of the hexagon is labeled O. There are points labeled at each vertex and at the midpoints of each side of the hexagon. Starting at the top of the image, the midpoint of the side is labeled F. Moving clockwise, the points are labeled G, H, I, J, K, L, M, N, N, P Q, and R. There are 6 dotted lines drawn through the hexagon and through point O. One connects midpoints F and L. One connects vertices G and M. One connects midpoints H and N. One connects vertices I and P. One connects midpoints J and Q. One connects vertices K and R.
What is the angle of rotation about O that maps PQ to RF?
(1 point)
Responses
240º
240º
210º
210º
90º
90º
300º
300º
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The angle of rotation about O that maps PQ to RF is 240º.
To find the angle of rotation that maps PQ to RF, we need to find the difference in positions of P and F when rotating about O.
1. Identify the positions of P and F in the original hexagon:
- P is a vertex of the hexagon.
- F is the midpoint of the side.
2. Count the number of vertices and midpoints from P to F:
- There are 3 vertices (P, Q, and R) and 3 midpoints (J, K, and L) between P and F.
3. Determine the total angle of the hexagon at O:
- Since the hexagon is regular, each angle at O is 360º/6 = 60º.
4. Calculate the angle between each vertex and midpoint:
- Since there are 3 vertices and 3 midpoints between P and F, each angle is (360º - 60º)/6 = 300º/6 = 50º.
5. Determine the angle of rotation needed to map PQ to RF:
- To go from vertex P to midpoints F, J, K, and L, we need to rotate 4 times.
- The total angle of rotation is 4 * 50º = 200º.
Therefore, the angle of rotation about O that maps PQ to RF is 200º.