The area of an isosceles triangle having base `2` `cm` and the length of one of the equal sides `4` `cm` is

Options

A. `sqrt15 cm^(2)`
B. `sqrt((15)/(2))cm^(2)`
C. `2sqrt15cm^(2)`
D. `4sqrt15cm^(2)`

Explanation

To find the area of the isosceles triangle with a base of 2 cm and equal sides of 4 cm, we can use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
We have:
- Length of the equal sides (a and b) = 4 cm
- Length of the base (c) = 2 cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{4 + 4 + 2}{2} = \frac{10}{2} = 5 \text{ cm} \]

### Step 3: Apply Heron's formula
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
Substituting the values we have:
\[ A = \sqrt{5(5 - 4)(5 - 4)(5 - 2)} \]
Calculating each term:
- \( s - a = 5 - 4 = 1 \)
- \( s - b = 5 - 4 = 1 \)
- \( s - c = 5 - 2 = 3 \)

So we can rewrite the area as:
\[ A = \sqrt{5 \times 1 \times 1 \times 3} \]
\[ A = \sqrt{15} \text{ cm}^2 \]

### Final Answer
The area of the isosceles triangle is \( \sqrt{15} \text{ cm}^2 \).

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