Question 1164863: Given a parallelogram ABCD, side AB is extended to a point P such that
AB : BP = 2 : 1 and side CD is extended to a point Q such that D is the
midpoint of CQ. The line PQ meets the diagonal AC at Z and the sides
BC and AD at X and Y , respectively.
(a) Find the ratio of the areas of triangles AZY and CZX.
(b) Find the ratio of the lengths PX and ZY .
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! 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?
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