PQRS is a trapezium with PQ y SR. If M and N are two points on the non-parallel 
sides PS and QR respectively, such that MN is parallel to PQ, then show that 
PM/MS
= 
QN/NR .

Let's analyze the given problem and provide a clear geometric proof for the statement.

**Problem Restatement:**
- We have a trapezium \( PQRS \) where \( PQ \parallel SR \).
- Points \( M \) and \( N \) lie on the non-parallel sides \( PS \) and \( QR \) respectively.
- \( MN \) is parallel to \( PQ \).

We need to show that:

\[
\frac{PM}{MS} = \frac{QN}{NR}
\]

### Proof:

Since \( PQ \parallel MN \parallel SR \), and \( MN \) is parallel to both bases \( PQ \) and \( SR \), the triangles \( \triangle PMN \) and \( \triangle QNR \) are similar to the triangles \( \triangle PQR \).

#### Step 1: Similar Triangles
- From the similarity of triangles:
  \[
  \triangle PMN \sim \triangle PSR \quad \text{(because corresponding angles are equal)}
  \]
  \[
  \triangle QNR \sim \triangle QSR \quad \text{(by the same reasoning)}
  \]

#### Step 2: Ratio of Corresponding Sides
From the similarity of triangles, we have:

\[
\frac{PM}{PS} = \frac{MN}{SR} \quad \text{and} \quad \frac{QN}{QR} = \frac{MN}{SR}
\]

Since both ratios \( \frac{PM}{PS} \) and \( \frac{QN}{QR} \) equal \( \frac{MN}{SR} \), it follows that:

\[
\frac{PM}{PS} = \frac{QN}{QR}
\]

#### Step 3: Expressing in Terms of Required Ratio
Let's rewrite \( PS \) and \( QR \) in terms of segments:
- \( PS = PM + MS \)
- \( QR = QN + NR \)

Now, using the above expressions:

\[
\frac{PM}{PM + MS} = \frac{QN}{QN + NR}
\]

Cross-multiplying, we get:

\[
PM \cdot (QN + NR) = QN \cdot (PM + MS)
\]

Expanding both sides:

\[
PM \cdot QN + PM \cdot NR = QN \cdot PM + QN \cdot MS
\]

Simplifying, we find that:

\[
PM \cdot NR = QN \cdot MS
\]

Finally, dividing both sides by \( MS \cdot NR \):

\[
\frac{PM}{MS} = \frac{QN}{NR}
\]

Thus, we have proved the required relation.

Would you like more details on any step, or have any questions? Here are some related questions to explore:

1. How can this proof be adapted if the trapezium is isosceles?
2. What if \( MN \) is not parallel to \( PQ \)? How does the ratio change?
3. How would this proof change if \( M \) and \( N \) were outside of the segments \( PS \) and \( QR \)?
4. Can you show a real-life example where this geometric property might be applied?
5. What would happen if points \( M \) and \( N \) were chosen such that \( MN \) was not parallel to either \( PQ \) or \( SR \)?

**Tip:** Similar triangles are a powerful tool for solving problems involving parallel lines and proportional segments.

PQRS is a trapezium with PQ parallel to SR. If M and N are two points on the non-parallel sides PS and QR respectively, such that MN is parallel to PQ, then show that PM/MS = QN/NR.

Learn how to prove that PM/MS = QN/NR in a trapezium PQRS with parallel sides PQ and SR. This detailed solution uses the properties of similar triangles and proportionality, suitable for students in Grades 9-12.

Math Problem Statement

PQRS is a trapezium with PQ y SR. If M and N are two points on the non-parallel sides PS and QR respectively, such that MN is parallel to PQ, then show that PM/MS

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