ABC is a right angled triangle, B being the right angle. Mid-points of BC and AC are respectively B' and A'. The ratio of the area of the quadrilateral AA' B'B to the area of the triangles ABC is

Options

A. `1:2`
B. `2:3`
C. `3:4`
D. None of the above

Answer

To solve the problem, we need to find the ratio of the area of the quadrilateral \( AA'B'B \) to the area of triangle \( ABC \). Let's break it down step by step.

### Step 1: Understand the Geometry
We have a right-angled triangle \( ABC \) with \( B \) as the right angle. The midpoints of sides \( BC \) and \( AC \) are denoted as \( B' \) and \( A' \) respectively.

### Step 2: Identify Areas
1. **Area of Triangle \( ABC \)**:
The area of triangle \( ABC \) can be calculated using the formula:
\[ \text{Area}_{ABC} = \frac{1}{2} \times AB \times BC \]
where \( AB \) and \( BC \) are the lengths of the two legs of the triangle.

2. **Area of Quadrilateral \( AA'B'B \)**:
To find the area of quadrilateral \( AA'B'B \), we can use the fact that it is formed by subtracting the area of triangle \( A'B'C \) from the area of triangle \( ABC \).

### Step 3: Calculate Area of Triangle \( A'B'C \)
Since \( A' \) and \( B' \) are midpoints, the lengths \( A'B \) and \( B'C \) will be half of \( AB \) and \( BC \) respectively. Therefore, the area of triangle \( A'B'C \) can be calculated as:
\[ \text{Area}_{A'B'C} = \frac{1}{2} \times A'B \times B'C = \frac{1}{2} \times \frac{1}{2} AB \times \frac{1}{2} BC = \frac{1}{4} \times \frac{1}{2} AB \times BC = \frac{1}{4} \text{Area}_{ABC} \]

### Step 4: Area of Quadrilateral \( AA'B'B \)
Now, we can find the area of quadrilateral \( AA'B'B \):
\[ \text{Area}_{AA'B'B} = \text{Area}_{ABC} - \text{Area}_{A'B'C} \]
Substituting the area of triangle \( A'B'C \):
\[ \text{Area}_{AA'B'B} = \text{Area}_{ABC} - \frac{1}{4} \text{Area}_{ABC} = \frac{3}{4} \text{Area}_{ABC} \]

### Step 5: Ratio of Areas
Now we can find the ratio of the area of quadrilateral \( AA'B'B \) to the area of triangle \( ABC \):
\[ \text{Ratio} = \frac{\text{Area}_{AA'B'B}}{\text{Area}_{ABC}} = \frac{\frac{3}{4} \text{Area}_{ABC}}{\text{Area}_{ABC}} = \frac{3}{4} \]

### Conclusion
Thus, the ratio of the area of quadrilateral \( AA'B'B \) to the area of triangle \( ABC \) is \( 3:4 \).