Find the area of an isosceles triangle which has base 12 cm and equal sides are 10 cm each.

Answer

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

### Step 1: Identify the sides of the triangle
The sides of the isosceles triangle are:
- Base (C) = 12 cm
- Equal sides (A and B) = 10 cm each

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

So, we can rewrite the area formula:
\[ A = \sqrt{16 \times 6 \times 6 \times 4} \]

### Step 4: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{16 \times 6 \times 6 \times 4} \]
Calculating step-by-step:
- \( 16 \times 4 = 64 \)
- \( 6 \times 6 = 36 \)
- Therefore, \( A = \sqrt{64 \times 36} \)

### Step 5: Calculate the square root
Now, we can find the square root:
\[ A = \sqrt{64} \times \sqrt{36} = 8 \times 6 = 48 \text{ cm}^2 \]

### Conclusion
The area of the isosceles triangle is \( 48 \text{ cm}^2 \).

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