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

Options

A. `4.5 : 8`
B. `4:3`
C. `8:4.5`
D. `3:4`

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 1 (ABC) and Triangle 2 (PQR). Both triangles have equal vertical angles (let's denote this angle as θ). The areas of these triangles are given to be in the ratio of 9:16.

2. **Using the Area Formula**:
The area \( A \) of an isosceles triangle can be expressed as:
\[ A = \frac{1}{2} \times b \times h \]
where \( b \) is the base and \( h \) is the height of the triangle.

3. **Setting Up the Ratio of Areas**:
Let the area of Triangle 1 (ABC) be \( A_1 \) and the area of Triangle 2 (PQR) be \( A_2 \). According to the problem:
\[ \frac{A_1}{A_2} = \frac{9}{16} \]

4. **Relating the Areas to Heights**:
Since both triangles have the same vertical angle θ, we can use the property that the ratio of the areas of two triangles with the same angle is equal to the product of the bases and heights. Therefore, we can express the areas in terms of their bases and heights:
\[ A_1 = \frac{1}{2} \times b_1 \times h_1 \]
\[ A_2 = \frac{1}{2} \times b_2 \times h_2 \]
Thus, we have:
\[ \frac{A_1}{A_2} = \frac{b_1 \times h_1}{b_2 \times h_2} \]

5. **Using the Ratio of Areas**:
From the area ratio, we can write:
\[ \frac{b_1 \times h_1}{b_2 \times h_2} = \frac{9}{16} \]

6. **Finding the Ratio of Heights**:
Since the triangles are isosceles and have equal angles, the bases \( b_1 \) and \( b_2 \) are proportional to the heights \( h_1 \) and \( h_2 \). Therefore, we can express the ratio of heights as:
\[ \frac{h_1}{h_2} = \frac{A_1 \cdot b_2}{A_2 \cdot b_1} \]
However, since we know the areas are in the ratio of 9:16, we can simplify this to:
\[ \frac{h_1}{h_2} = \sqrt{\frac{A_1}{A_2}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

7. **Final Answer**:
The ratio of the corresponding heights of the two triangles is:
\[ \frac{h_1}{h_2} = \frac{3}{4} \]