Find the area of triangle whose sides are 9 cm, 12cm and 15 cm. Also, find the length of altitude corresponding to the largest side of the triangle.

Answer

To find the area of the triangle with sides 9 cm, 12 cm, and 15 cm, and the length of the altitude corresponding to the largest side, follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 9 \, \text{cm} \)
- \( b = 12 \, \text{cm} \)
- \( c = 15 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated as:
\[ s = \frac{a + b + c}{2} = \frac{9 + 12 + 15}{2} = \frac{36}{2} = 18 \, \text{cm} \]

### Step 3: Use Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
Substituting the values:
\[ A = \sqrt{18(18 - 9)(18 - 12)(18 - 15)} \]
Calculating each term:
\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]

### Step 4: Simplify the expression
Now, simplify:
\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]
We can factor out the numbers:
\[ = \sqrt{(2 \times 9) \times 9 \times (2 \times 3) \times 3} \]
Grouping the pairs:
\[ = \sqrt{(2^2) \times (9^2) \times (3^2)} = 2 \times 9 \times 3 = 54 \, \text{cm}^2 \]

### Step 5: Find the altitude corresponding to the largest side
The largest side \( c \) is 15 cm. The area can also be expressed using the base and height:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Let the height corresponding to the base (15 cm) be \( h \):
\[ 54 = \frac{1}{2} \times 15 \times h \]
Multiplying both sides by 2:
\[ 108 = 15h \]
Dividing both sides by 15:
\[ h = \frac{108}{15} = 7.2 \, \text{cm} \]

### Final Results
- The area of the triangle is \( 54 \, \text{cm}^2 \).
- The length of the altitude corresponding to the largest side is \( 7.2 \, \text{cm} \).

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