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

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.