If area of two similar triangle are equal then ratio of their corresponding altitude is.

Options

A. `1:1`
B. `1:sqrt(2)`
C. `1:2`
D. Cannot be determined

Answer

To solve the problem, we need to find the ratio of the corresponding altitudes of two similar triangles when their areas are equal. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
- We have two similar triangles, let's denote them as triangle ABC and triangle PQR.
- Since the triangles are similar, the ratio of their corresponding sides is constant.

2. **Area of Similar Triangles**:
- The area of similar triangles is proportional to the square of the ratio of their corresponding sides.
- Mathematically, if the ratio of the sides of triangle ABC to triangle PQR is \( k \), then:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = k^2 \]

3. **Given Condition**:
- We are given that the areas of the two triangles are equal:
\[ \text{Area of } \triangle ABC = \text{Area of } \triangle PQR \]
- This implies:
\[ k^2 = 1 \quad \Rightarrow \quad k = 1 \]
- Therefore, the ratio of the corresponding sides is 1:1.

4. **Finding the Ratio of Altitudes**:
- The area of a triangle can also be expressed in terms of its base and height (altitude). For triangle ABC, the area can be written as:
\[ \text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height} \]
- Let the base be \( BC \) and the height (altitude) be \( AD \).
- For triangle PQR, the area can be expressed as:
\[ \text{Area of } \triangle PQR = \frac{1}{2} \times \text{base} \times \text{height} \]
- Let the base be \( QR \) and the height (altitude) be \( PS \).

5. **Equating the Areas**:
- Since the areas are equal, we can set up the equation:
\[ \frac{1}{2} \times BC \times AD = \frac{1}{2} \times QR \times PS \]
- Canceling \( \frac{1}{2} \) from both sides gives:
\[ BC \times AD = QR \times PS \]

6. **Using the Ratio of Bases**:
- From our earlier conclusion, we found that \( BC = QR \) (since the ratio of corresponding sides is 1:1).
- Substituting this into the equation gives:
\[ BC \times AD = BC \times PS \]
- Dividing both sides by \( BC \) (assuming \( BC \neq 0 \)) results in:
\[ AD = PS \]

7. **Final Ratio of Altitudes**:
- Therefore, the ratio of the altitudes \( AD \) and \( PS \) is:
\[ \frac{AD}{PS} = 1 \]
- This means the ratio of their corresponding altitudes is 1:1.

### Conclusion:
The ratio of the corresponding altitudes of the two similar triangles, when their areas are equal, is 1:1.