Find the area and perimeter of a triangle whose sides are 17 cm , 8 cm and 15 cm long.

Answer

To find the area and perimeter of a triangle with sides measuring 17 cm, 8 cm, and 15 cm, we can follow these steps:

### Step 1: Calculate the Perimeter
The perimeter \( P \) of a triangle is the sum of the lengths of its sides.

\[ P = A + B + C \]
Where:
- \( A = 17 \, \text{cm} \)
- \( B = 8 \, \text{cm} \)
- \( C = 15 \, \text{cm} \)

Calculating the perimeter:

\[ P = 17 + 8 + 15 = 40 \, \text{cm} \]

### Step 2: Calculate the Semi-Perimeter
The semi-perimeter \( S \) is half of the perimeter.

\[ S = \frac{P}{2} = \frac{40}{2} = 20 \, \text{cm} \]

### Step 3: Apply 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:

\[ A = \sqrt{20(20 - 17)(20 - 8)(20 - 15)} \]

Calculating each term:

- \( S - A = 20 - 17 = 3 \)
- \( S - B = 20 - 8 = 12 \)
- \( S - C = 20 - 15 = 5 \)

Now substituting these values back into the formula:

\[ A = \sqrt{20 \times 3 \times 12 \times 5} \]

### Step 4: Calculate the Product Inside the Square Root
Calculating the product:

\[ 20 \times 3 = 60 \]
\[ 60 \times 12 = 720 \]
\[ 720 \times 5 = 3600 \]

### Step 5: Calculate the Square Root
Now, we find the square root of 3600:

\[ A = \sqrt{3600} = 60 \, \text{cm}^2 \]

### Final Results
- **Perimeter**: \( 40 \, \text{cm} \)
- **Area**: \( 60 \, \text{cm}^2 \)

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