Find the area of triangle whose sides are 17 cm, 8 cm and 15 cm. Also 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, and to calculate the length of the altitude corresponding to the largest side, we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- Side a = 17 cm (largest side)
- Side b = 15 cm
- Side c = 8 cm

### Step 2: Verify if the triangle is a right triangle
To check if the triangle is a right triangle, we can use the Pythagorean theorem, which states that for a right triangle, the square of the hypotenuse (largest side) is equal to the sum of the squares of the other two sides.

\[ a^2 = b^2 + c^2 \]

Substituting the values:
\[ 17^2 = 15^2 + 8^2 \]
Calculating each square:
\[ 289 = 225 + 64 \]
\[ 289 = 289 \]

Since both sides are equal, the triangle is a right triangle.

### Step 3: Calculate the area of the triangle
The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Here, we can take the base as 15 cm and the height as 8 cm:
\[ A = \frac{1}{2} \times 15 \times 8 \]
Calculating the area:
\[ A = \frac{1}{2} \times 120 = 60 \text{ cm}^2 \]

### Step 4: Calculate the altitude corresponding to the largest side
The altitude \( H \) corresponding to the largest side (17 cm) can be calculated using the area we found:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is 17 cm and the height is \( H \):
\[ 60 = \frac{1}{2} \times 17 \times H \]
Multiplying both sides by 2:
\[ 120 = 17H \]
Now, solving for \( H \):
\[ H = \frac{120}{17} \approx 7.06 \text{ cm} \]

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