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 of 9:16.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two isosceles triangles, let's call them Triangle ABC and Triangle DEF. Both triangles have equal vertical angles (∠A = ∠D). The areas of these triangles are given in the ratio of 9:16.

2. **Using the Area Ratio**:
The area of a triangle can be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Since both triangles have the same vertical angle, we can use the formula for the area in terms of their bases and heights.

3. **Setting Up the Ratio of Areas**:
Let the heights of Triangle ABC and Triangle DEF be \(h_1\) and \(h_2\) respectively, and let their bases be \(b_1\) and \(b_2\). The areas can be expressed as:
\[ \text{Area of Triangle ABC} = \frac{1}{2} b_1 h_1 \]
\[ \text{Area of Triangle DEF} = \frac{1}{2} b_2 h_2 \]
Given the ratio of the areas:
\[ \frac{\frac{1}{2} b_1 h_1}{\frac{1}{2} b_2 h_2} = \frac{9}{16} \]
This simplifies to:
\[ \frac{b_1 h_1}{b_2 h_2} = \frac{9}{16} \]

4. **Using the Property of Similar Triangles**:
Since the triangles are isosceles and have equal vertical angles, they are similar. Therefore, the ratio of their corresponding sides (bases) is equal to the ratio of their corresponding heights:
\[ \frac{b_1}{b_2} = \frac{h_1}{h_2} \]
Let \(k = \frac{h_1}{h_2}\). Then we can express \(b_1\) in terms of \(b_2\):
\[ b_1 = k \cdot b_2 \]

5. **Substituting Back into the Area Ratio**:
Substitute \(b_1\) into the area ratio:
\[ \frac{k \cdot b_2 \cdot h_1}{b_2 \cdot h_2} = \frac{9}{16} \]
This simplifies to:
\[ k \cdot \frac{h_1}{h_2} = \frac{9}{16} \]
Since \(k = \frac{h_1}{h_2}\), we can substitute:
\[ k^2 = \frac{9}{16} \]

6. **Finding the Height Ratio**:
Taking the square root of both sides:
\[ k = \frac{3}{4} \]
Therefore, the ratio of the heights \(h_1\) to \(h_2\) is:
\[ \frac{h_1}{h_2} = \frac{3}{4} \]

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