Corresponding sides of two triangles are in the ratio `2\ :3` . If the area of the smaller triangle is `48\ c m^2` , determine the area of the larger triangle.

Options

A. `106 cm^2`
B. `110 cm^2`
C. `108 cm^2`
D. `104 cm^2`

Explanation

To determine the area of the larger triangle given that the corresponding sides of two triangles are in the ratio of 2:3 and the area of the smaller triangle is 48 cm², we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Ratio of Corresponding Sides**:
The sides of the smaller triangle (let's call it Triangle ABC) and the larger triangle (Triangle DEF) are in the ratio 2:3. This means:
\[ \frac{AB}{DE} = \frac{2}{3} \]

2. **Understand the Relationship of Areas**:
When two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore:
\[ \frac{\text{Area of Triangle ABC}}{\text{Area of Triangle DEF}} = \left(\frac{AB}{DE}\right)^2 \]

3. **Substitute Known Values**:
We know the area of Triangle ABC (the smaller triangle) is 48 cm². Let the area of Triangle DEF (the larger triangle) be \( X \). Thus, we can write:
\[ \frac{48}{X} = \left(\frac{2}{3}\right)^2 \]

4. **Calculate the Square of the Ratio**:
Calculate \( \left(\frac{2}{3}\right)^2 \):
\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

5. **Set Up the Equation**:
Now we can substitute this back into our equation:
\[ \frac{48}{X} = \frac{4}{9} \]

6. **Cross-Multiply to Solve for \( X \)**:
Cross-multiplying gives us:
\[ 48 \cdot 9 = 4 \cdot X \]
Simplifying this, we get:
\[ 432 = 4X \]

7. **Isolate \( X \)**:
Divide both sides by 4 to find \( X \):
\[ X = \frac{432}{4} = 108 \]

8. **Conclusion**:
Therefore, the area of the larger triangle (Triangle DEF) is:
\[ \text{Area of Triangle DEF} = 108 \, \text{cm}^2 \]

### Final Answer:
The area of the larger triangle is **108 cm²**.