Question: User: H.7 Rotate polygons about a point Xv
Look at this diagram:
G
H
F
Which diagram shows $\triangle FGH$ rotated $115^\circ$ counterclockwise about $G$?
A top-left graph: the green triangle has points $G$ at the left, $H$ at the upper right, and $F$ at the lower center; the blue image shares $G$ and has points $H'$ at the lower right and $F'$ at the lower left; position_hint \in \{top-left\}
B top-right graph: the green triangle has $G$ at the left, $H$ at the upper right, and $F$ at the lower center; the blue image has $H'$ at the upper left and $F'$ at the right; position_hint \in \{top-right\}
C center graph: the green triangle has $G$ at the left, $H$ at the right, and $F$ at the lower center; the blue image has $H'$ at the upper left and $F'$ at the upper right; position_hint \in \{center\}
D bottom-left graph: the green triangle has $G$ at the left, $H$ at the right, and $F$ at the lower center; the blue image has $F'$ at the upper left and $H'$ at the upper right; position_hint \in \{bottom-left\}
1. **Problem Statement:** We need to determine which diagram shows $\triangle FGH$ rotated $115^\circ$ counterclockwise about point $G$.
2. **Understanding Rotation:** Rotation about a point means every point of the figure moves around that point by the given angle in the specified direction (counterclockwise here).
3. **Key Point:** Since the rotation is about $G$, point $G$ remains fixed in all diagrams.
4. **Rotation Angle:** $115^\circ$ counterclockwise means each point $H$ and $F$ moves around $G$ by $115^\circ$ in the counterclockwise direction.
5. **Visualizing the Rotation:**
- Original $H$ is at upper right from $G$.
- Rotating $H$ $115^\circ$ counterclockwise from upper right will place $H'$ roughly to the upper left or left side.
- Original $F$ is at lower center from $G$.
- Rotating $F$ $115^\circ$ counterclockwise will move $F'$ to somewhere upper left or upper right depending on exact positions.
6. **Checking Each Diagram:**
- **A (top-left):** $H'$ is at lower right, $F'$ at lower left — this is clockwise rotation, so incorrect.
- **B (top-right):** $H'$ at upper left, $F'$ at right — $F'$ should move upward, so $F'$ at right is unlikely.
- **C (center):** $H'$ at upper left, $F'$ at upper right — both points moved counterclockwise from original positions, consistent with $115^\circ$ rotation.
- **D (bottom-left):** $F'$ at upper left, $H'$ at upper right — order of points swapped, inconsistent with rotation preserving orientation.
7. **Conclusion:** Diagram **C** correctly shows $\triangle FGH$ rotated $115^\circ$ counterclockwise about $G$.
**Final answer:** Diagram C.