Find the area of an isosceles triangle each of whose equal sides measures 13 cm and whose base measures 20 cm.

Options

A. 53.06 `cm^(2)`
B. 63.06 `cm^(2)`
C. 83.06 `cm^(2)`
D. 73.06 `cm^(2)`

Answer

To find the area of the isosceles triangle with equal sides measuring 13 cm and a base measuring 20 cm, we will use Heron's formula. Here are the steps:

### Step 1: Identify the sides of the triangle
Let the lengths of the sides of the triangle be:
- \( a = 13 \, \text{cm} \) (one of the equal sides)
- \( b = 13 \, \text{cm} \) (the other equal side)
- \( c = 20 \, \text{cm} \) (the base)

### 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:
\[ s = \frac{13 + 13 + 20}{2} = \frac{46}{2} = 23 \, \text{cm} \]

### 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:
\[ A = \sqrt{23 \times (23 - 13) \times (23 - 13) \times (23 - 20)} \]
Calculating the individual terms:
- \( s - a = 23 - 13 = 10 \)
- \( s - b = 23 - 13 = 10 \)
- \( s - c = 23 - 20 = 3 \)

Now substituting these values back into the formula:
\[ A = \sqrt{23 \times 10 \times 10 \times 3} \]

### Step 4: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{23 \times 10^2 \times 3} = \sqrt{23 \times 100 \times 3} = \sqrt{6900} \]

### Step 5: Calculate the square root
To find the area:
\[ A \approx \sqrt{6900} \approx 83.06 \, \text{cm}^2 \]

### Final Answer
The area of the isosceles triangle is approximately \( 83.06 \, \text{cm}^2 \).

---