https://file.aiquickdraw.com/mathbot/file/1729186109226xp4lhwdc.jpg

Let's go through the questions one by one based on the image you provided:

### 6(a) - Describe fully the single transformation that maps shape A onto shape B.
The transformation here is a **rotation**. Shape A is rotated **90 degrees clockwise** around the origin \((0, 0)\) to map onto shape B.

### 6(b) - Describe fully the single transformation that maps shape A onto shape C.
This transformation is a **translation**. To map shape A onto shape C, we need to move it 7 units to the right and 9 units upwards. The translation vector is therefore \(\begin{pmatrix}7 \\ 9\end{pmatrix}\).

### 6(c) - On the grid, draw the image of shape A after a translation by the vector \(\begin{pmatrix}3 \\ 7\end{pmatrix}\).
To translate shape A by \(\begin{pmatrix}3 \\ 7\end{pmatrix}\), add 3 units to each x-coordinate and 7 units to each y-coordinate of the vertices of shape A.

### 6(d) - On the grid, draw the image of shape B after a reflection in the line \(y = 1\).
To reflect shape B in the line \(y = 1\), the distance of each point of shape B from the line \(y = 1\) must be reversed across the line. Simply mirror the points of shape B across the line \(y = 1\).

### 7(a) - Work out \(\begin{pmatrix}-\frac{6}{5}\end{pmatrix} + \begin{pmatrix}\frac{8}{-1}\end{pmatrix}\).
Let's calculate it step-by-step:
\[
-\frac{6}{5} + \frac{8}{-1} = -\frac{6}{5} - 8 = -\frac{6}{5} - \frac{40}{5} = -\frac{46}{5}.
\]

### 7(b) - Work out \(3 \times \begin{pmatrix}-\frac{4}{7}\end{pmatrix}\).
This is a simple multiplication:
\[
3 \times -\frac{4}{7} = -\frac{12}{7}.
\]

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Let me know if you need further clarifications or a detailed step-by-step for any of these!

### Relative Questions:
1. What are the steps to determine the center of rotation for two shapes?
2. How would the transformation change if shape A was reflected instead of rotated in 6(a)?
3. Can a combination of transformations be simplified into a single transformation? If yes, how?
4. What is the rule for translating shapes along both the x and y axes?
5. How do we determine the equation of the line of reflection in geometric transformations?

### Tip:
Always check whether the transformation is rigid (like rotation, translation, and reflection) to ensure that the shapes maintain their size and form after the transformation.

Describe the transformations and work out the vector and fraction problems as seen in the image provided.

Learn how to describe geometric transformations such as rotation, reflection, and translation with step-by-step solutions to IGCSE mathematics problems. This guide also includes vector arithmetic and fraction calculations for students in Grades 9-10.

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