https://file.aiquickdraw.com/mathbot/file/1726921200359yryh8vmw.jpg

Let's break down the question in the image step by step. The given diagram shows a trapezium \( PQRS \), and you are asked to express \( \overrightarrow{PR} \) in terms of \( x \) and \( y \). Here's what is provided:

### Given information:
- \( \overrightarrow{PQ} = 8x \)
- \( \overrightarrow{PS} = 24y \)
- \( \overrightarrow{PT} = \frac{2}{3} \overrightarrow{PS} \)
- \( \overrightarrow{QR} = \frac{3}{4} \overrightarrow{PS} \)

### Required:
To express \( \overrightarrow{PR} \) in terms of \( x \) and \( y \).

### Solution:
1. Start by recalling that the vector \( \overrightarrow{PR} \) can be written as:
   \[
   \overrightarrow{PR} = \overrightarrow{PQ} + \overrightarrow{QR}
   \]
2. You are given \( \overrightarrow{PQ} = 8x \) and \( \overrightarrow{QR} = \frac{3}{4} \overrightarrow{PS} \). Since \( \overrightarrow{PS} = 24y \), we substitute:
   \[
   \overrightarrow{QR} = \frac{3}{4} \times 24y = 18y
   \]
3. Now, substituting both vectors into the expression for \( \overrightarrow{PR} \):
   \[
   \overrightarrow{PR} = 8x + 18y
   \]

Thus, \( \overrightarrow{PR} = 8x + 18y \).

Would you like more detailed steps or explanations? Here are five related questions to extend the discussion:

1. How do you calculate the magnitude of vector \( \overrightarrow{PR} \)?
2. What is the direction of \( \overrightarrow{PT} \) with respect to \( \overrightarrow{PS} \)?
3. How do you express vector \( \overrightarrow{QS} \) in terms of \( x \) and \( y \)?
4. If you are given the coordinates of points \( P \) and \( R \), how can you find \( \overrightarrow{PR} \)?
5. How would the solution change if \( \overrightarrow{QR} \) was expressed in terms of a different fraction of \( \overrightarrow{PS} \)?

### Tip:
When working with vectors, always pay attention to their directions and whether they are scaled versions of other vectors.

Express PR in terms of x and y based on the trapezium PQRS, where PQ = 8x, PS = 24y, PT = (2/3) PS, and QR = (3/4) PS.

This problem involves expressing the vector PR in terms of x and y for trapezium PQRS, given PQ = 8x, PS = 24y, PT = (2/3) PS, and QR = (3/4) PS. Learn how to apply vector addition to solve for PR.

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