Find the area of an isosceles triangle of sides 10 cm , 10 cm , and 12 cm .

Answer

To find the area of an isosceles triangle with sides 10 cm, 10 cm, and 12 cm, we can use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
The sides of the triangle are:
- A = 10 cm
- B = 10 cm
- C = 12 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:
\[ s = \frac{10 + 10 + 12}{2} = \frac{32}{2} = 16 \text{ cm} \]

### Step 3: Apply Heron's formula for the area
Heron's formula for the area (A) of a triangle is given by:
\[ A = \sqrt{s \cdot (s - A) \cdot (s - B) \cdot (s - C)} \]
Substituting the values we have:
\[ A = \sqrt{16 \cdot (16 - 10) \cdot (16 - 10) \cdot (16 - 12)} \]
Calculating each term:
- \( s - A = 16 - 10 = 6 \)
- \( s - B = 16 - 10 = 6 \)
- \( s - C = 16 - 12 = 4 \)

So we have:
\[ A = \sqrt{16 \cdot 6 \cdot 6 \cdot 4} \]

### Step 4: Simplify the expression
Calculating the product:
\[ A = \sqrt{16 \cdot 36 \cdot 4} \]
We can break this down:
\[ A = \sqrt{(4^2) \cdot (6^2) \cdot (2^2)} \]
This simplifies to:
\[ A = \sqrt{(4 \cdot 6 \cdot 2)^2} = 4 \cdot 6 \cdot 2 \]

### Step 5: Calculate the final area
Calculating the final value:
\[ A = 48 \text{ cm}^2 \]

### Final Answer
The area of the isosceles triangle is **48 cm²**.

---