Vertical angles of two isoceles triangles are equal. Then corresponding altitudes are in the ratio `4:9`. Find the ratio of their areas :

Options

A. `16:49`
B. `16:81`
C. `16:65`
D. none of these

Answer

To solve the problem, we need to find the ratio of the areas of two isosceles triangles given that their vertical angles are equal and the corresponding altitudes are in the ratio of 4:9.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have two isosceles triangles, let's denote them as Triangle ABC and Triangle PQR.
- The vertical angles (angle A in Triangle ABC and angle P in Triangle PQR) are equal.
- The corresponding altitudes (AD for Triangle ABC and PO for Triangle PQR) are in the ratio 4:9.

2. **Setting Up the Ratio of Altitudes**:
- Given: \( \frac{AD}{PO} = \frac{4}{9} \)

3. **Finding the Ratio of the Areas**:
- The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Since the vertical angles are equal and both triangles are isosceles, the triangles are similar. Therefore, the ratio of their areas is proportional to the square of the ratio of their corresponding heights (altitudes).
- Thus, we can write:
\[ \frac{\text{Area of Triangle ABC}}{\text{Area of Triangle PQR}} = \left(\frac{AD}{PO}\right)^2 \]

4. **Calculating the Ratio of Areas**:
- Substituting the ratio of the altitudes:
\[ \frac{\text{Area of Triangle ABC}}{\text{Area of Triangle PQR}} = \left(\frac{4}{9}\right)^2 \]
- Calculating the square:
\[ \left(\frac{4}{9}\right)^2 = \frac{16}{81} \]

5. **Final Ratio of Areas**:
- Therefore, the ratio of the areas of Triangle ABC to Triangle PQR is:
\[ \text{Area of Triangle ABC} : \text{Area of Triangle PQR} = 16 : 81 \]

### Conclusion:
The ratio of the areas of the two isosceles triangles is \( 16 : 81 \).