To solve this problem, we need to determine the new coordinates of the vertices $F$, $G$, and $H$ after a 180° clockwise rotation around the point $(1, -2)$.

Steps to Perform a 180° Rotation

For a 180° rotation about a specific point $(a, b)$, the rotation formula for any point $(x, y)$ is: $(x', y') = (2a - x, 2b - y)$ This formula shifts the coordinates of each point to its rotated location.

Given Points and Rotation Point

• Rotation Point $(a, b) = (1, -2)$
• Points to rotate:
  • $F(2, -1)$
  • $G(5, -3)$
  • $H(1, -4)$

Applying the Formula

1. For $F(2, -1)$: $F' = (2 \cdot 1 - 2, 2 \cdot -2 - (-1))$ $F' = (2 - 2, -4 + 1) = (0, -3)$

2. For $G(5, -3)$: $G' = (2 \cdot 1 - 5, 2 \cdot -2 - (-3))$ $G' = (2 - 5, -4 + 3) = (-3, -1)$

3. For $H(1, -4)$: $H' = (2 \cdot 1 - 1, 2 \cdot -2 - (-4))$ $H' = (2 - 1, -4 + 4) = (1, 0)$

Result

After a 180° clockwise rotation about the point $(1, -2)$, the vertices of the triangle become:

• $F' = (0, -3)$
• $G' = (-3, -1)$
• $H' = (1, 0)$

Would you like further explanation on any part of this solution, or have additional questions?

Here are five related questions to expand on this concept:

1. How would the coordinates change if the rotation was 90° instead of 180°?
2. What would the coordinates be if we rotated the triangle around the origin instead of $(1, -2)$?
3. Can you apply this rotation formula to other shapes, like rectangles or polygons?
4. How can you verify the results of a rotation mathematically?
5. What is the geometric significance of rotating by 180°?

Tip: When rotating points around a specific center, remember that a 180° rotation simply reflects each point through that center, making it an easy transformation to visualize.