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

Options

A. `75 cm^2`
B. `60 cm^2`
C. `45 cm^2`
D. `120 cm^2`

Explanation

To find the area of a triangle with sides measuring 17 cm, 8 cm, and 15 cm using Heron's formula, follow these steps:

### Step 1: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of a triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.

In this case:
- \( a = 17 \) cm
- \( b = 8 \) cm
- \( c = 15 \) cm

Calculating \( s \):
\[ s = \frac{17 + 8 + 15}{2} = \frac{40}{2} = 20 \text{ cm} \]

### Step 2: Use Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]

Now, substitute the values:
\[ A = \sqrt{20(20 - 17)(20 - 8)(20 - 15)} \]

Calculating each term:
- \( s - a = 20 - 17 = 3 \)
- \( s - b = 20 - 8 = 12 \)
- \( s - c = 20 - 15 = 5 \)

So, substituting these values back into the formula:
\[ A = \sqrt{20 \times 3 \times 12 \times 5} \]

### Step 3: Simplify the expression
Now, calculate the product inside the square root:
\[ 20 \times 3 = 60 \]
\[ 60 \times 12 = 720 \]
\[ 720 \times 5 = 3600 \]

Thus, we have:
\[ A = \sqrt{3600} \]

### Step 4: Calculate the square root
Now, calculate the square root:
\[ A = 60 \text{ cm}^2 \]

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

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