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### **Problem 8**  
\( y \) is directly proportional to the square root of \( (x - 3) \).

This gives the equation:
\[
y = k \sqrt{x - 3}
\]
where \( k \) is the constant of proportionality.

**Step 1:** Find \( k \) using the given values \( x = 28 \) and \( y = 20 \).

\[
20 = k \sqrt{28 - 3} = k \sqrt{25} = k \times 5
\]
\[
k = \frac{20}{5} = 4
\]

**Step 2:** Now that we know \( k = 4 \), substitute it into the equation:
\[
y = 4 \sqrt{x - 3}
\]

**Step 3:** Find \( y \) when \( x = 39 \).

\[
y = 4 \sqrt{39 - 3} = 4 \sqrt{36} = 4 \times 6 = 24
\]

**Answer:** When \( x = 39 \), \( y = 24 \).

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### **Problem 9a**  
Describe the **single transformation**:

#### (i) Triangle A to Triangle B  
This is a **rotation**.  
- Rotation: 90° clockwise (or -90°)  
- Centre of rotation: the origin \( (0, 0) \)

#### (ii) Triangle A to Triangle C  
This is a **reflection**.  
- Mirror line: the **y-axis** \( x = 0 \)

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### **Problem 9b**  
Perform a **translation** on triangle A by the vector:
\[
\begin{pmatrix} 2 \\ 10 \end{pmatrix}
\]
This means each point of triangle A moves **2 units right** and **10 units up**.

If you need to draw this on the grid:
- Add \( 2 \) to the \( x \)-coordinates of each vertex of triangle A.
- Add \( 10 \) to the \( y \)-coordinates of each vertex.

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Let me know if you have further questions!

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### **Related Questions**  
1. How do you find the constant of proportionality for other types of proportional relationships?
2. What are the steps for identifying the type of rotation given the center and angle?
3. How do you determine the mirror line for different types of reflections?
4. Can you describe the difference between translation and rotation transformations?
5. What is the process for checking if a transformation was correctly applied?

**Tip:** Always double-check proportionality problems by verifying units and constants before solving for unknowns.

y is directly proportional to the square root of (x - 3). When x = 28, y = 20. Find y when x = 39. Also, describe the transformation of triangle A onto triangles B and C, and perform a translation of triangle A by the vector (2, 10).

Solve a problem involving direct proportionality where y is proportional to the square root of (x - 3). Additionally, explore geometric transformations, including rotations, reflections, and translations, mapping triangle A onto triangles B and C. Suitable for Grades 9-10.

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