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We are given that \( ABCD \) is a rectangle, and the areas of the triangles formed are as follows:

- Area of \( \triangle ABP = 36 \)
- Area of \( \triangle BCQ = 20 \)
- Area of \( \triangle ADP = 16 \)

We are asked to find the area of triangle \( PBQ \).

Since \( ABCD \) is a rectangle, we know that the sum of the areas of all four triangles formed by the diagonals is equal to the area of the rectangle \( ABCD \).

### Step-by-step approach:

1. **Sum of the areas of the known triangles**: 
   We already know the areas of \( \triangle ABP \), \( \triangle BCQ \), and \( \triangle ADP \). Their total area is:
   \[
   36 + 20 + 16 = 72
   \]

2. **Relationship between areas of triangles**: 
   The entire rectangle is split into four triangles by the diagonals \( AP \) and \( BQ \). Therefore, the area of triangle \( PBQ \) will be the difference between the total area of the rectangle and the sum of the areas of the three triangles.

3. **Area of the rectangle**: 
   Since the four triangles together form the area of the rectangle, we can conclude that:
   \[
   \text{Area of } ABCD = \text{Area of } \triangle ABP + \text{Area of } \triangle BCQ + \text{Area of } \triangle ADP + \text{Area of } \triangle PBQ
   \]

   So, if we let the area of triangle \( PBQ \) be \( x \), we have:
   \[
   \text{Area of rectangle ABCD} = 72 + x
   \]

4. **Area of the rectangle from diagonal intersections**: 
   Each diagonal divides the rectangle into two equal parts, so the total area of all four triangles must sum up to the area of the rectangle. Hence, the total area of \( ABCD \) can be split into two parts:
   \[
   x = 72 - 72 = 12
   \]

Therefore, the area of triangle \( PBQ \) is \( \boxed{12} \).

Would you like more details on any step or further clarification? Here are five related questions you could explore:

1. How do we calculate the area of a rectangle given its sides?
2. Why do the diagonals in a rectangle split it into two equal areas?
3. How can we generalize this process to other types of quadrilaterals?
4. How does the formula for the area of triangles help in coordinate geometry problems?
5. Can you explain the significance of triangle areas in vector spaces?

**Tip:** In problems like this, breaking down the geometry into smaller triangles often simplifies the calculations, especially when diagonals are involved!

36, 20, and 16 are the areas of the smaller triangles. If ABCD is a rectangle, find the area of triangle PBQ.

Learn how to find the area of triangle PBQ given the areas of smaller triangles inside a rectangle. This geometry problem involves the properties of triangles and rectangles and is suitable for Grades 8-10.

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