Let's break down the transformations needed to map triangle $A$ onto triangles $B$ and $C$.

Part (i) Mapping Triangle A onto Triangle B

• Transformation Type: This appears to be a rotation.
• Details: We need to check the position of triangle $B$ relative to triangle $A$. Triangle $B$ is on the left side of the $y$-axis, and triangle $A$ is on the right side. Looking at the structure and orientation of both triangles, triangle $B$ seems to be a 180-degree rotation of triangle $A$ about the origin $(0, 0)$.

Thus, the transformation is:

• Rotation of 180° about the origin.

Part (ii) Mapping Triangle A onto Triangle C

• Transformation Type: This looks like a translation.
• Details: Triangle $C$ is directly below triangle $A$, and the shape and orientation are preserved, indicating no rotation or reflection. The shift is purely vertical and horizontal.

The translation vector from $A$ to $C$ can be determined by calculating the movement from a corresponding point on triangle $A$ to triangle $C$.

• Translation vector: $\begin{pmatrix} -4 \\ -8 \end{pmatrix}$.

Therefore, the transformation is:

• Translation by the vector $\begin{pmatrix} -4 \\ -8 \end{pmatrix}$.

Would you like more details on these transformations or any other questions related to them?

Here are five relative questions to deepen your understanding:

1. How can we verify the center of rotation in transformations?
2. What are the general steps to identify a reflection between two figures?
3. Could you calculate the area of triangles $A$, $B$, and $C$?
4. How can we distinguish between different types of transformations visually?
5. What are the coordinates of the vertices for triangle $A$ after performing the described transformations?

Tip: Always plot a few key points when working with transformations to ensure accuracy in your descriptions.