One of the equal sides of an isosceles triangle is 13 cm and its perimeter is 50 cm. Find the area of the triangle.

Answer

To solve the problem of finding the area of an isosceles triangle where one of the equal sides is 13 cm and the perimeter is 50 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the two equal sides of the isosceles triangle be \( a = 13 \) cm each. Let the base of the triangle be \( b \). The perimeter of the triangle is given by:
\[ \text{Perimeter} = a + a + b = 50 \text{ cm} \]
Substituting the values, we have:
\[ 13 + 13 + b = 50 \]

### Step 2: Solve for the base \( b \)
Combine the equal sides:
\[ 26 + b = 50 \]
Now, subtract 26 from both sides:
\[ b = 50 - 26 = 24 \text{ cm} \]

### Step 3: Calculate the semi-perimeter \( s \)
The semi-perimeter \( s \) is half of the perimeter:
\[ s = \frac{\text{Perimeter}}{2} = \frac{50}{2} = 25 \text{ cm} \]

### Step 4: Use Heron's formula to find the area
Heron's formula states that the area \( A \) of a triangle can be calculated using the semi-perimeter and the lengths of the sides:
\[ A = \sqrt{s(s-a)(s-a)(s-b)} \]
Substituting the values:
\[ A = \sqrt{25(25-13)(25-13)(25-24)} \]
Calculating each term:
\[ A = \sqrt{25 \times 12 \times 12 \times 1} \]

### Step 5: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{25 \times 144} = \sqrt{3600} \]

### Step 6: Calculate the area
Taking the square root:
\[ A = 60 \text{ cm}^2 \]

### Final Answer
The area of the triangle is \( 60 \text{ cm}^2 \).

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