The area of ∆ABC is 40. Points P, Q and R lie on sides AB, BC and CA respectively. If AP = 3 and PB = 5, and the area of AABQ is equal to the area of PBQR, determine the area of AQC.

Given the problem, we need to determine the area of triangle \( AQC \) based on the provided information. Let's break it down step by step:

### Step 1: Analyze the Triangle \( \triangle ABC \)
The total area of \( \triangle ABC \) is given as 40 square units.

### Step 2: Use the Information Given for Points P, Q, and R
- Points \( P \), \( Q \), and \( R \) lie on sides \( AB \), \( BC \), and \( CA \), respectively.
- \( AP = 3 \) and \( PB = 5 \). So, the ratio of \( AP \) to \( PB \) is \( \frac{AP}{PB} = \frac{3}{5} \).

### Step 3: Apply the Area Ratio of \( \triangle ABQ \) and Quadrilateral \( PBQR \)
We are given that the area of \( \triangle ABQ \) is equal to the area of quadrilateral \( PBQR \).

### Step 4: Area of \( \triangle ABQ \)
First, let's calculate the area of \( \triangle ABQ \).

The ratio \( \frac{AP}{PB} = \frac{3}{5} \) implies that the area of \( \triangle APQ \) (which is part of \( \triangle ABQ \)) is also \( \frac{3}{8} \) of the area of \( \triangle ABQ \).

Let's denote the area of \( \triangle ABQ \) as \( x \). Then:

\[
\text{Area of } \triangle APQ = \frac{3}{8} \times x
\]
\[
\text{Area of } \triangle PBQ = \frac{5}{8} \times x
\]

Since the total area of \( \triangle ABQ \) equals the area of quadrilateral \( PBQR \), we can express the total area of \( \triangle ABQ \) as \( x \).

### Step 5: Express the Area of \( \triangle ABC \)
The total area of \( \triangle ABC \) is:

\[
\text{Area of } \triangle ABC = \text{Area of } \triangle ABQ + \text{Area of } \triangle AQC = 40
\]

Since \( \triangle ABQ \) equals the area of quadrilateral \( PBQR \), and we have:

\[
\text{Area of } \triangle ABC = x + \text{Area of } \triangle AQC
\]

### Step 6: Solve for \( \text{Area of } \triangle AQC \)
The area of \( \triangle ABQ \) is half the area of \( \triangle ABC \), given that the area of \( \triangle ABQ \) equals the area of \( PBQR \):

\[
x = 20
\]

Therefore, the area of \( \triangle AQC \) is:

\[
\text{Area of } \triangle AQC = 40 - x = 40 - 20 = 20 \text{ square units}
\]

### Final Answer
The area of \( \triangle AQC \) is 20 square units.

Do you have any questions or want more details?

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**Tip:** Always remember that the area of a triangle can be split proportionally if a line segment is parallel to one of its sides.

Learn how to determine the area of triangle AQC using given ratios and areas in triangle ABC. This problem involves calculating triangle areas based on point positions and area ratios. Suitable for high school students studying geometry.

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