Find the area of an isosceles triangle whose equal sides are 8 cm each and the third side is 10 cm ?

Options

A. `10 cm^(2)`
B. `48 cm^(2)`
C. `5sqrt(39)cm^(2)`
D. `10 sqrt(10) cm^(2)`

Answer

To find the area of an isosceles triangle with equal sides of 8 cm and a base of 10 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
- Let the lengths of the equal sides (A and B) be 8 cm each.
- Let the length of the base (C) be 10 cm.

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter (s) of a triangle is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{8 + 8 + 10}{2} = \frac{26}{2} = 13 \text{ cm} \]

### Step 3: Use Heron's formula to find the area
Heron's formula for the area (A) of a triangle is given by:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values we have:
\[ A = \sqrt{13(13 - 8)(13 - 8)(13 - 10)} \]
Calculating each term:
- \(s - A = 13 - 8 = 5\)
- \(s - B = 13 - 8 = 5\)
- \(s - C = 13 - 10 = 3\)

Now substituting these values into the formula:
\[ A = \sqrt{13 \times 5 \times 5 \times 3} \]

### Step 4: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{13 \times 5^2 \times 3} = \sqrt{13 \times 25 \times 3} = \sqrt{975} \]

### Step 5: Further simplify \(\sqrt{975}\)
We can simplify \(\sqrt{975}\) as follows:
\[ 975 = 25 \times 39 \]
Thus,
\[ \sqrt{975} = \sqrt{25 \times 39} = \sqrt{25} \times \sqrt{39} = 5\sqrt{39} \]

### Final Answer
The area of the isosceles triangle is:
\[ \text{Area} = 5\sqrt{39} \text{ cm}^2 \]

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