ABCD is a rectangle. P & Q are two points on AB, such that AP : PQ : QB = 3 : 4 : 5. Find the ratio of area `DeltaPQC` and rectangle ABCD.

Answer

To solve the problem, we need to find the ratio of the area of triangle PQC to the area of rectangle ABCD. Let's break it down step by step.

### Step 1: Understand the Ratios
We are given that the segments AP, PQ, and QB are in the ratio 3:4:5. Let's denote:
- AP = 3x
- PQ = 4x
- QB = 5x

### Step 2: Calculate the Length of AB
The total length of AB can be calculated by adding the lengths of AP, PQ, and QB:
\[ AB = AP + PQ + QB = 3x + 4x + 5x = 12x \]

### Step 3: Identify the Height of the Triangle and Rectangle
Let the height of the rectangle (and triangle) from line AB to line CD be denoted as h (which is the same for both the triangle and rectangle).

### Step 4: Calculate the Area of Triangle PQC
The base of triangle PQC is PQ, which is 4x. The height is h. Therefore, the area of triangle PQC can be calculated using the formula for the area of a triangle:
\[ \text{Area of } \triangle PQC = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4x \times h = 2xh \]

### Step 5: Calculate the Area of Rectangle ABCD
The area of rectangle ABCD is given by the formula:
\[ \text{Area of rectangle ABCD} = \text{length} \times \text{height} = AB \times h = 12x \times h = 12xh \]

### Step 6: Find the Ratio of the Areas
Now we can find the ratio of the area of triangle PQC to the area of rectangle ABCD:
\[ \text{Ratio} = \frac{\text{Area of } \triangle PQC}{\text{Area of rectangle ABCD}} = \frac{2xh}{12xh} \]
Here, \(xh\) cancels out:
\[ \text{Ratio} = \frac{2}{12} = \frac{1}{6} \]

### Final Answer
The ratio of the area of triangle PQC to the area of rectangle ABCD is:
\[ \text{Ratio} = 1 : 6 \]