Write True or False and justify your answer:
The area of the isosceles triangle is `(5)/(4)sqrt11` `cm^(2)` if the perimeter is `11` `cm` and the base is `5` `cm`.

Explanation

To determine whether the statement about the area of the isosceles triangle is true or false, we will follow these steps:

### Step 1: Identify the given values
- Perimeter (P) = 11 cm
- Base (a) = 5 cm
- Area (A) = \(\frac{5}{4}\sqrt{11}\) cm²

### Step 2: Calculate the lengths of the equal sides
Since the triangle is isosceles, we denote the lengths of the two equal sides as \(b\). The perimeter of the triangle can be expressed as:
\[ P = b + b + a = 2b + a \]
Substituting the known values:
\[ 11 = 2b + 5 \]
Now, we solve for \(b\):
\[ 2b = 11 - 5 \]
\[ 2b = 6 \]
\[ b = \frac{6}{2} = 3 \text{ cm} \]

### Step 3: Calculate the area using Heron's Formula
To use Heron's formula, we first need to calculate the semi-perimeter (s):
\[ s = \frac{P}{2} = \frac{11}{2} = 5.5 \text{ cm} \]

Now, we can apply Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-b)} \]
Substituting the values:
\[ A = \sqrt{5.5(5.5 - 5)(5.5 - 3)(5.5 - 3)} \]
Calculating each term:
- \(s - a = 5.5 - 5 = 0.5\)
- \(s - b = 5.5 - 3 = 2.5\)

Now substituting these into the formula:
\[ A = \sqrt{5.5 \times 0.5 \times 2.5 \times 2.5} \]
Calculating:
\[ A = \sqrt{5.5 \times 0.5 \times 6.25} \]
\[ A = \sqrt{5.5 \times 3.125} \]
\[ A = \sqrt{17.1875} \]

### Step 4: Compare the calculated area with the given area
Now we need to determine if:
\[ \sqrt{17.1875} \text{ is equal to } \frac{5}{4}\sqrt{11} \]

Calculating \(\frac{5}{4}\sqrt{11}\):
- First, calculate \(\sqrt{11} \approx 3.3166\)
- Then, \(\frac{5}{4} \times 3.3166 \approx 4.14575\)

Now, squaring both sides to compare:
- \(17.1875\) (from our calculation)
- \((\frac{5}{4}\sqrt{11})^2 = \frac{25}{16} \times 11 = \frac{275}{16} \approx 17.1875\)

### Conclusion
Since both areas match, the statement is **True**.