Let's solve each transformation for triangle $\Delta A$ based on the provided instructions and diagram.

Triangle $\Delta A$'s initial position:

• The coordinates of triangle $\Delta A$ appear to be approximately $(5, 2)$, $(7, 2)$, and $(6, 4)$.

1. Reflection over $y = -1$:

• A reflection over $y = -1$ inverts points across the line $y = -1$. The vertical distance from each point of $\Delta A$ to the line $y = -1$ is calculated and mirrored across.
• The result should give the image as $\Delta C$ (the triangle on the left, below the x-axis).

2. Rotation $180^\circ$ about the origin:

• A $180^\circ$ rotation inverts all points through the origin. The transformation rule is $(x, y) \to (-x, -y)$.
• Applying this to each vertex of $\Delta A$, the new image lands at triangle $\Delta E$ (located in the bottom-right quadrant).

3. Translation by $(x + 3, y + 2)$:

• Each vertex of $\Delta A$ is moved 3 units right and 2 units up.
• After applying this, triangle $\Delta A$ maps to $\Delta G$ (the triangle near the top right).

4. Reflection over $x = 5$:

• A reflection over the vertical line $x = 5$ flips points horizontally across $x = 5$. The distances from each point of $\Delta A$ to the line are measured and inverted.
• This transformation results in triangle $\Delta D$ (the triangle to the top-left of $\Delta A$).

Summary of transformations:

• $\Delta A$ reflected over $y = -1$ → $\Delta C$
• $\Delta A$ rotated $180^\circ$ about the origin → $\Delta E$
• $\Delta A$ translated by $(x + 3, y + 2)$ → $\Delta G$
• $\Delta A$ reflected over $x = 5$ → $\Delta D$

Would you like further details on any of these transformations?

Here are 5 related questions you might explore:

1. How do you determine the reflection of a point across any given line?
2. What is the effect of a $90^\circ$ rotation compared to a $180^\circ$ rotation?
3. How does translating a figure affect its orientation and size?
4. Can reflections across non-horizontal or non-vertical lines be calculated similarly?
5. How can you identify rotational symmetries in other shapes, like polygons?

Tip: To verify transformations, it can help to sketch out the original and transformed figures step by step on graph paper!