To find the ratio of the areas of two similar triangles when the ratio of their corresponding sides is given, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Ratio of Corresponding Sides:**
The problem states that the corresponding sides of the two similar triangles are in the ratio of 3:4. Let's denote the sides of triangle A as \( a_1, a_2, a_3 \) and the sides of triangle B as \( b_1, b_2, b_3 \). Therefore, we have:
\[
\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} = \frac{3}{4}
\]
2. **Use the Property of Similar Triangles:**
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. If the ratio of the sides is \( \frac{3}{4} \), then the ratio of the areas \( \frac{Area \, of \, Triangle \, A}{Area \, of \, Triangle \, B} \) can be expressed as:
\[
\frac{Area \, of \, Triangle \, A}{Area \, of \, Triangle \, B} = \left(\frac{3}{4}\right)^2
\]
3. **Calculate the Square of the Ratio:**
Now, we calculate the square of \( \frac{3}{4} \):
\[
\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}
\]
4. **State the Ratio of the Areas:**
Thus, the ratio of the areas of triangle A to triangle B is:
\[
\frac{Area \, of \, Triangle \, A}{Area \, of \, Triangle \, B} = \frac{9}{16}
\]
5. **Final Answer:**
Therefore, the ratio of the areas of the two similar triangles is \( 9:16 \).
