Find the area of a triangle :
whose three sides are 17 cm, 8 cm and 15 cm long.
Also, in part (ii) of the question, calculate the length of the altitude corresponding to the largest side of the triangle.

Answer

To find the area of the triangle with sides 17 cm, 8 cm, and 15 cm, we will use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
Let:
- Side A = 8 cm
- Side B = 17 cm
- Side C = 15 cm

### 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:
\[ S = \frac{8 + 17 + 15}{2} = \frac{40}{2} = 20 \text{ cm} \]

### Step 3: Apply Heron's formula to find the area (A)
Heron's formula states:
\[ \text{Area} = \sqrt{S \times (S - A) \times (S - B) \times (S - C)} \]
Substituting the values:
\[ \text{Area} = \sqrt{20 \times (20 - 8) \times (20 - 17) \times (20 - 15)} \]
Calculating each term:
\[ = \sqrt{20 \times 12 \times 3 \times 5} \]

### Step 4: Simplify the expression
Calculating the product:
\[ = \sqrt{20 \times 12 \times 3 \times 5} \]
Breaking it down:
\[ = \sqrt{(2 \times 2 \times 5) \times (2 \times 2 \times 3) \times 3 \times 5} \]
\[ = \sqrt{(2^4) \times (3^2) \times (5^2)} \]
Taking the square root:
\[ = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 \text{ cm}^2 \]

### Step 5: Calculate the altitude corresponding to the largest side
The largest side is 17 cm. The area can also be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Let \( h \) be the height corresponding to the base of 17 cm:
\[ 60 = \frac{1}{2} \times 17 \times h \]
Rearranging gives:
\[ h = \frac{60 \times 2}{17} = \frac{120}{17} \approx 7.06 \text{ cm} \]

### Final Answers
1. The area of the triangle is \( 60 \text{ cm}^2 \).
2. The length of the altitude corresponding to the largest side is approximately \( 7.06 \text{ cm} \).

---