Find the area and perimeter of an isosceles triangle whose equal sides are 5 cm each and base is 6 cm .

Answer

To find the area and perimeter of an isosceles triangle with equal sides of 5 cm each and a base of 6 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the triangle be labeled as ABC, where:
- AB = AC = 5 cm (the equal sides)
- BC = 6 cm (the base)

### Step 2: Calculate the perimeter
The perimeter \( P \) of a triangle is the sum of the lengths of all its sides. Therefore, we can calculate the perimeter as follows:
\[ P = AB + AC + BC \]
Substituting the values:
\[ P = 5 \, \text{cm} + 5 \, \text{cm} + 6 \, \text{cm} = 16 \, \text{cm} \]

### Step 3: Calculate the semi-perimeter
The semi-perimeter \( S \) is half of the perimeter. We can calculate it as:
\[ S = \frac{P}{2} = \frac{16 \, \text{cm}}{2} = 8 \, \text{cm} \]

### Step 4: Use Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{S(S - AB)(S - AC)(S - BC)} \]
Substituting the known values:
\[ A = \sqrt{8 \, \text{cm} \times (8 \, \text{cm} - 5 \, \text{cm}) \times (8 \, \text{cm} - 5 \, \text{cm}) \times (8 \, \text{cm} - 6 \, \text{cm})} \]
Calculating each term inside the parentheses:
\[ A = \sqrt{8 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} \times 2 \, \text{cm}} \]
Now, calculate the product:
\[ A = \sqrt{8 \times 3 \times 3 \times 2} = \sqrt{144} = 12 \, \text{cm}^2 \]

### Final Results
- **Perimeter**: \( 16 \, \text{cm} \)
- **Area**: \( 12 \, \text{cm}^2 \)