The perimeter of an isosceles triangle is 32 cm. The ratio of one of the equal sides to its base is 3:2. Find the area of the triangle.

Options

A. `48cm^(2)`
B. `28 sqrt3 cm^(2)`
C. `32 sqrt2 cm^(2)`
D. `44cm^(2)`

Answer

To find the area of the isosceles triangle given the perimeter and the ratio of its sides, we can follow these steps:

### Step 1: Define the sides of the triangle
Let the equal sides of the isosceles triangle be represented as \(3R\) each, and the base as \(2R\).

### Step 2: Set up the perimeter equation
The perimeter of the triangle is given as 32 cm. Therefore, we can write the equation:
\[ 3R + 3R + 2R = 32 \]
This simplifies to:
\[ 8R = 32 \]

### Step 3: Solve for \(R\)
Now, divide both sides by 8 to find \(R\):
\[ R = \frac{32}{8} = 4 \]

### Step 4: Calculate the lengths of the sides
Now that we have \(R\), we can find the lengths of the sides:
- Equal sides: \(3R = 3 \times 4 = 12\) cm
- Base: \(2R = 2 \times 4 = 8\) cm

### Step 5: Calculate the semi-perimeter
The semi-perimeter \(s\) is half of the perimeter:
\[ s = \frac{32}{2} = 16 \text{ cm} \]

### Step 6: Apply Heron's formula to find the area
Heron's formula for the area \(A\) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \(a\), \(b\), and \(c\) are the sides of the triangle. Here, \(a = 12\), \(b = 12\), and \(c = 8\).

Substituting the values into Heron's formula:
\[ A = \sqrt{16(16-12)(16-12)(16-8)} \]
Calculating each term:
- \(s - a = 16 - 12 = 4\)
- \(s - b = 16 - 12 = 4\)
- \(s - c = 16 - 8 = 8\)

Now substituting these values:
\[ A = \sqrt{16 \times 4 \times 4 \times 8} \]

### Step 7: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{16 \times 4 \times 4 \times 8} = \sqrt{2048} \]

### Step 8: Further simplify \(\sqrt{2048}\)
We can factor \(2048\) into \(1024 \times 2\):
\[ A = \sqrt{1024 \times 2} = \sqrt{1024} \times \sqrt{2} = 32\sqrt{2} \text{ cm}^2 \]

### Final Answer
Thus, the area of the isosceles triangle is:
\[ \text{Area} = 32\sqrt{2} \text{ cm}^2 \]