The area of a rhombus is equal to the area of a triangle. If base of triangle is 24 cm, its corresponding altitude is 16 cm and one of the diagonals of the rhombus is 19.2 cm, find its other diagonal.

Answer

To solve the problem, we need to find the other diagonal of the rhombus given that the area of the rhombus is equal to the area of a triangle. Here are the steps to find the solution:

### Step-by-Step Solution:

1. **Calculate the Area of the Triangle**:
- The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Given the base of the triangle is 24 cm and the height (altitude) is 16 cm, we can substitute these values into the formula:
\[ \text{Area} = \frac{1}{2} \times 24 \times 16 \]
- Performing the multiplication:
\[ \text{Area} = \frac{1}{2} \times 384 = 192 \text{ cm}^2 \]

2. **Set the Area of the Rhombus Equal to the Area of the Triangle**:
- We know the area of the rhombus is equal to the area of the triangle, which we calculated to be 192 cm².

3. **Use the Formula for the Area of a Rhombus**:
- The area of a rhombus can also be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
- Here, \(d_1\) is one diagonal, and \(d_2\) is the other diagonal. We know \(d_1 = 19.2\) cm.

4. **Substitute the Known Values into the Area Formula**:
- We can set up the equation:
\[ \frac{1}{2} \times 19.2 \times d_2 = 192 \]

5. **Solve for the Other Diagonal \(d_2\)**:
- First, multiply both sides by 2 to eliminate the fraction:
\[ 19.2 \times d_2 = 384 \]
- Now, divide both sides by 19.2 to solve for \(d_2\):
\[ d_2 = \frac{384}{19.2} \]
- Performing the division:
\[ d_2 = 20 \text{ cm} \]

6. **Conclusion**:
- The other diagonal of the rhombus is 20 cm.