Question 1164863
To solve this problem, we will use the properties of similar triangles and the ratios given for the extensions of the parallelogram sides.

### 1. Set up the Coordinates and Ratios

Let the vertices of the parallelogram be , , , and .
The vector  has length .

* **Point P:** Since  and  is an extension of , . The total distance .
* **Point Q:** Since  is the midpoint of , then . In a parallelogram, . Therefore,  is an extension of  such that , making .

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### (a) Ratio of the Areas of  and 

First, we find the positions of  and  using similar triangles.
Since  and , the line  creates several pairs of similar triangles.

1. **Find  on :** .
* The ratio of their bases is . Since  is the midpoint of ,  and .
* However,  is an extension of  past .  is not a single line segment; we look at the vertical heights.
* Using the transversal , we find that  divides  such that .


2. **Find  on :**  (where  is the line  extended).
* .
*  (along the base line) is  in terms of horizontal shift.
* By similar triangles,  divides  such that .


3. **Find  on :**
Since  is not immediately obvious, we look at the segments.  is on  and  is on . Because ,  is indeed similar to .
* The ratio of similarity  is the ratio of  to .
* From our ratios:  and .
* Since  (opposite sides of a parallelogram):


* The ratio of the **areas** of similar triangles is the square of the ratio of their corresponding lengths:





**Answer (a):** The ratio of the areas of  and  is **144 : 25**.

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### (b) Ratio of the lengths  and 

To find this ratio, we use the Intercept Theorem (Thales's Theorem) along the line .

Let the total length of the line segment  be divided by the intersection points .
Using the ratios derived from the similar triangles:

1. ** relative to :** From  (where  is the extension line), .
So, .
2. ** relative to :** By considering the intersection with the diagonal , we find .
3. ** relative to :** From , we find .

Calculating the segment :



Substituting the fractional values of :


Now, find the ratio :


**Answer (b):** The ratio of the lengths  is **44 : 7**.

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Would you like me to provide a coordinate geometry proof for these specific intersection points to verify the ratios?