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To determine the coordinates of the vertices of triangle \( \triangle F'G'H' \) after a \(180^\circ\) clockwise rotation of \( \triangle FGH \) about the point \( (1, -3) \), we need to apply the rotation transformation to each vertex \( F(-5, -4) \), \( G(-3, -2) \), and \( H(1, -5) \) with respect to the center of rotation \( (1, -3) \).

### Steps for a \(180^\circ\) Rotation:
For a \(180^\circ\) rotation about a point \( (a, b) \), the transformed coordinates \( (x', y') \) of any point \( (x, y) \) can be found using:
\[
(x', y') = (2a - x, 2b - y)
\]
where \( (a, b) = (1, -3) \) in this case.

### Calculations for Each Vertex:

1. **For \( F(-5, -4) \):**
   \[
   F' = (2 \cdot 1 - (-5), 2 \cdot (-3) - (-4))
   \]
   \[
   F' = (2 + 5, -6 + 4) = (7, -2)
   \]

2. **For \( G(-3, -2) \):**
   \[
   G' = (2 \cdot 1 - (-3), 2 \cdot (-3) - (-2))
   \]
   \[
   G' = (2 + 3, -6 + 2) = (5, -4)
   \]

3. **For \( H(1, -5) \):**
   \[
   H' = (2 \cdot 1 - 1, 2 \cdot (-3) - (-5))
   \]
   \[
   H' = (2 - 1, -6 + 5) = (1, -1)
   \]

### Final Result:
The vertices of \( \triangle F'G'H' \) after the \(180^\circ\) rotation are:
- \( F'(7, -2) \)
- \( G'(5, -4) \)
- \( H'(1, -1) \)

Would you like further details on the rotation process, or have any questions on transformations?

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Here are 5 questions related to this topic for further exploration:
1. How would a \(90^\circ\) clockwise rotation affect the coordinates of the vertices?
2. What would the coordinates be after a \(270^\circ\) clockwise rotation?
3. How can we use matrices to represent these transformations?
4. What if the center of rotation was a different point, such as the origin?
5. How can we determine if a rotation preserves the shape and size of a figure?

**Tip:** For rotations around a point other than the origin, always translate to the origin first, apply the rotation, and then translate back.

Triangle FGH is shown below and has vertices F(-5, -4), G(-3, -2), and H(1, -5). After a 180° clockwise rotation of FGH about the point (1, -3), the result is ΔF'G'H'.

Learn how to find the coordinates of a triangle's vertices after a 180° clockwise rotation around a specific point. This example demonstrates transforming triangle FGH with vertices F(-5, -4), G(-3, -2), and H(1, -5) using coordinate geometry and transformation formulas. Suitable for high school students studying rotations and transformations.

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