Two isosceles triangles have equal vertical angles and their areas are in the ratio 9:16 . Find the ratio of their corresponding heights.

Answer

To solve the problem, we need to find the ratio of the corresponding heights of two isosceles triangles that have equal vertical angles and whose areas are in the ratio 9:16.

### Step-by-Step Solution:

1. **Understanding the Isosceles Triangles**:
Let triangle ABC and triangle PQR be the two isosceles triangles. In triangle ABC, let AB = AC, and in triangle PQR, let PQ = PR. Since both triangles have equal vertical angles, we can denote the angles as follows:
- For triangle ABC: ∠A = ∠P (the equal vertical angles)
- For triangle PQR: ∠B = ∠C and ∠Q = ∠R (the base angles)

2. **Using the Area Ratio**:
We are given that the areas of triangle ABC and triangle PQR are in the ratio of 9:16. This can be expressed mathematically as:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{9}{16} \]

3. **Relating Area to Height**:
The area \( A \) of a triangle can be expressed in terms of its base \( b \) and height \( h \) as:
\[ A = \frac{1}{2} \times b \times h \]
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding heights. Therefore, we can write:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{h_1^2}{h_2^2} \]
where \( h_1 \) and \( h_2 \) are the heights from the vertices to the bases of triangles ABC and PQR respectively.

4. **Setting Up the Equation**:
From the area ratio, we have:
\[ \frac{h_1^2}{h_2^2} = \frac{9}{16} \]

5. **Taking the Square Root**:
To find the ratio of the heights, we take the square root of both sides:
\[ \frac{h_1}{h_2} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

6. **Conclusion**:
Therefore, the ratio of the corresponding heights of the two triangles is:
\[ h_1 : h_2 = 3 : 4 \]

### Final Answer:
The ratio of the corresponding heights of the two isosceles triangles is \( 3 : 4 \).