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The problem you've shared involves a rectangle \( ABCD \) and an extended point \( P \) on the side \( DC \). Let's break it down based on the details visible in the image:

- \( ABCD \) is a rectangle.
- \( P \) is a point on the side \( DC \) extended.
- \( Q \) is the midpoint of the line \( AP \).
- The area of the triangle \( BPQ \) is given as 20 square centimeters.
- We are asked to find the area of the rectangle \( ABCD \).

### Strategy:

1. **Set variables:**
   Let \( AB = h \) (the height of the rectangle) and \( BC = w \) (the width of the rectangle). Therefore, the area of the rectangle is \( A_{\text{rect}} = h \times w \).

2. **Calculate the area of triangle \( BPQ \):**
   The area of triangle \( BPQ \) is given as 20 cm². The area of a triangle is calculated using the formula:
   \[
   A_{\triangle} = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   Here, the base of the triangle \( BP \) would be along the extended side of the rectangle, and \( Q \), being the midpoint of \( AP \), adds complexity to finding its height.

3. **Relating triangle area to rectangle:**
   By analyzing the geometric properties and given data, we need to use proportions or other geometric relations to relate the area of triangle \( BPQ \) with the area of rectangle \( ABCD \).

I'll proceed with these calculations now.From the solution, we get that \( h = \frac{80}{w} \). This indicates a relationship between the height \( h \) and width \( w \) of the rectangle based on the given conditions and the area of the triangle.

To calculate the area of the rectangle, we find:
\[
\text{Area of the rectangle} = h \times w = \frac{80}{w} \times w = 80
\]
Thus, the area of the rectangle is **80 square centimeters**.

Would you like more details on the geometric reasoning or any clarification?

Here are 5 related questions:
1. How would the problem change if point \( P \) was inside the rectangle?
2. What if triangle \( BPQ \)'s area was given as a different value?
3. Can the area of triangle \( APQ \) help verify the result?
4. What geometric properties of midpoints help in problems like this?
5. How does extending a side affect calculations in coordinate geometry?

**Tip**: Always check if a midpoint divides the figure symmetrically to simplify calculations.

In the figure, ABCD is a rectangle, and P is a point on the side DC extended. Q is the midpoint of the line AP. The area of the triangle BPQ is 20 square centimeters. What is the area of the rectangle?

Solve a problem involving a rectangle and triangle BPQ, where the area of triangle BPQ is given as 20 square centimeters. Learn how to apply geometric concepts and formulas to calculate the area of the rectangle. Suitable for Grades 9-12 students.

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