The area of a rhombus is equal to the area of a triangle having base 24.8 cm and the corresponding height 16.5 cm. If one of the diagonals of the rhombus is 22 cm, find the length of the other diagonal.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the area of the triangle
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the base is 24.8 cm and the height is 16.5 cm, we can substitute these values into the formula.

\[ \text{Area} = \frac{1}{2} \times 24.8 \times 16.5 \]

### Step 2: Perform the multiplication
Calculating the multiplication:
\[ \text{Area} = \frac{1}{2} \times 24.8 \times 16.5 = \frac{1}{2} \times 409.2 = 204.6 \text{ cm}^2 \]

### Step 3: Set the area of the rhombus equal to the area of the triangle
Since the area of the rhombus is equal to the area of the triangle, we have:
\[ \text{Area of rhombus} = 204.6 \text{ cm}^2 \]

### Step 4: Use the formula for the area of a rhombus
The area of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
where \(d_1\) and \(d_2\) are the lengths of the diagonals. We know \(d_1 = 22 \text{ cm}\) and we need to find \(d_2\).

### Step 5: Substitute the known values into the area formula
Substituting the known values into the area formula:
\[ 204.6 = \frac{1}{2} \times 22 \times d_2 \]

### Step 6: Simplify the equation
To eliminate the fraction, we can multiply both sides by 2:
\[ 2 \times 204.6 = 22 \times d_2 \]
\[ 409.2 = 22 \times d_2 \]

### Step 7: Solve for \(d_2\)
Now, divide both sides by 22 to find \(d_2\):
\[ d_2 = \frac{409.2}{22} \]
Calculating this gives:
\[ d_2 = 18.6 \text{ cm} \]

### Final Answer
The length of the other diagonal \(d_2\) is **18.6 cm**.

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