Two isosceles triangles have equal angles and their areas are in the ratio `36 : 81` .The ratio of their corrersponding heights is

Options

A. ` 2 : 3`
B. ` 5 : 4`
C. ` 3 : 2`
D. `1 : 4`

Answer

To solve the problem, we need to find the ratio of the corresponding heights of two isosceles triangles given that their areas are in the ratio of 36:81.

### Step-by-Step Solution:

1. **Understanding the given information**:
We have two isosceles triangles, let's denote them as Triangle EBC and Triangle PQR. The areas of these triangles are given in the ratio of 36:81.

2. **Using the area ratio**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides (or heights). Therefore, if the areas of the triangles are in the ratio of \( \frac{36}{81} \), we can express this as:
\[ \frac{Area_{EBC}}{Area_{PQR}} = \frac{36}{81} \]

3. **Simplifying the area ratio**:
We can simplify the ratio \( \frac{36}{81} \):
\[ \frac{36}{81} = \frac{4}{9} \]

4. **Setting up the equation for heights**:
Let \( h_1 \) be the height of Triangle EBC and \( h_2 \) be the height of Triangle PQR. Since the triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding heights:
\[ \frac{Area_{EBC}}{Area_{PQR}} = \left(\frac{h_1}{h_2}\right)^2 \]
Therefore, we have:
\[ \frac{4}{9} = \left(\frac{h_1}{h_2}\right)^2 \]

5. **Taking the square root**:
To find the ratio of the heights, we take the square root of both sides:
\[ \frac{h_1}{h_2} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

6. **Conclusion**:
The ratio of the corresponding heights of the two isosceles triangles is:
\[ \frac{h_1}{h_2} = \frac{2}{3} \]