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The problem involves a trapezium PQRS, with some given vector relationships and a condition about the points \(P\), \(T\), and \(R\). Let's break it down.

### Information from the problem:
1. PQRS is a trapezium with \(PQ \parallel SR\).
2. \(\overrightarrow{SR} = \mathbf{a}\), \(\overrightarrow{PQ} = 3\mathbf{a}\), and \(\overrightarrow{PS} = \mathbf{b}\).
3. \(T\) is a point on \(SQ\) such that \(ST = \frac{1}{4}SQ\).
4. \(\overrightarrow{PT} = k \overrightarrow{PR}\), where \(k\) is a fraction.

### Key Steps:
1. **Understanding Vector Relationships:**
   - From the fact that \(T\) is a point on line \(SQ\), we know that it divides this line in a specific ratio. Since \(ST = \frac{1}{4}SQ\), the vector relationship can be expressed as:
     \[
     \overrightarrow{TQ} = \frac{3}{4} \overrightarrow{SQ}
     \]
   
2. **The Meaning of the Equation \(\overrightarrow{PT} = k \overrightarrow{PR}\):**
   - This equation implies that \(P\), \(T\), and \(R\) are **collinear**, because \(\overrightarrow{PT}\) is a scalar multiple (fraction \(k\)) of \(\overrightarrow{PR}\). When vectors between three points are proportional, it means these points lie on the same straight line.

### Conclusion:
The result tells us that points \(P\), \(T\), and \(R\) are **collinear**.

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Would you like more details on any specific part of the solution? Here are some questions to expand your understanding:

1. How can we derive the value of \(k\) from the given information?
2. What are the implications of \(PQ \parallel SR\) in terms of vector relationships?
3. How does the division ratio \(ST = \frac{1}{4} SQ\) affect the coordinates of point \(T\)?
4. Can the area of the trapezium PQRS be calculated using vectors?
5. How can we express the vector \(\overrightarrow{PQ}\) in terms of \(\overrightarrow{PS}\) and \(\overrightarrow{SR}\)?

**Tip:** When dealing with collinear points, the condition that a vector between two points is a scalar multiple of another vector is often key to recognizing this geometry.

PQRS is a trapezium with PQ parallel to SR. Vector relations: SR = a, PQ = 3a, PS = b. T is a point on SQ such that ST = (1/4)SQ. PT = kPR, where k is a fraction. What does this result tell you about the points P, T, and R?

Explore how to solve a problem involving vector relationships in a trapezium PQRS. Key concepts include vectors, collinearity, and line division ratios. Learn about the condition PT = kPR, where k is a fraction, and how it shows collinearity between points P, T, and R. Suitable for students in Grades 10-12.

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