We are given a rectangle $ABCD$ and several triangles within it, where the areas of three of the smaller triangles are labeled: 36, 20, and 16. We need to find the area of triangle $PBQ$.

Here is a breakdown of the key parts of the diagram:

• The areas of triangles $PAB$, $BQC$, and $PQD$ are 36, 20, and 16, respectively.
• The unknown area is for triangle $PBQ$, and we will need to use geometric properties and relationships within the rectangle.

Step-by-step approach:

1. Total Area of the Rectangle $ABCD$: The area of the rectangle can be found by adding up the areas of all the smaller triangles:
   $\text{Total Area} = 36 + 20 + 16 + \text{Area of } PBQ$ Let $A_{PBQ}$ be the area of triangle $PBQ$.

2. Geometric Relationships: The diagonals and segments dividing the rectangle, such as lines from $P$ to $B$ and from $P$ to $Q$, can help determine how the areas are distributed.

To solve this, we'll set up an equation based on the total area of the rectangle and the areas of the known triangles.

Let me calculate it step by step.The total area of the rectangle $ABCD$ is 72 square units.

Now, we can calculate the area of triangle $PBQ$ by subtracting the areas of the three given triangles from the total area of the rectangle:

$\text{Area of } PBQ = \text{Total Area} - (36 + 20 + 16)$

$\text{Area of } PBQ = 72 - (36 + 20 + 16) = 72 - 72 = 0$

Therefore, the area of triangle $PBQ$ is $0$, which suggests that point $P$, $B$, and $Q$ are collinear.

Would you like any further clarifications or details?

Here are some follow-up questions for deeper understanding:

1. How can we prove that $P$, $B$, and $Q$ are collinear?
2. What are the properties of collinear points in a geometric shape?
3. How do areas of triangles change with the position of points on a line?
4. Can we find the coordinates of points $P$, $B$, and $Q$ to confirm collinearity?
5. How does this result affect the overall geometry of rectangle $ABCD$?

Tip: When the area of a triangle is zero, it indicates that all three points of the triangle lie on the same straight line.