The sides of a triangle are in the ratio 13 : 14 : 15 and its perimeter is 84 cm . Find the area of the triangle .

Answer

To find the area of the triangle with sides in the ratio 13:14:15 and a perimeter of 84 cm, we can follow these steps:

### Step 1: Determine the sides of the triangle
Let the sides of the triangle be represented as:
- Side a = 13x
- Side b = 14x
- Side c = 15x

The perimeter of the triangle is given by the sum of its sides:
\[ a + b + c = 84 \]
Substituting the expressions for the sides:
\[ 13x + 14x + 15x = 84 \]
Combining like terms:
\[ 42x = 84 \]

### Step 2: Solve for x
To find the value of x, divide both sides by 42:
\[ x = \frac{84}{42} = 2 \]

### Step 3: Calculate the lengths of the sides
Now that we have the value of x, we can find the lengths of the sides:
- Side a = \( 13x = 13 \times 2 = 26 \) cm
- Side b = \( 14x = 14 \times 2 = 28 \) cm
- Side c = \( 15x = 15 \times 2 = 30 \) cm

### Step 4: Calculate the semi-perimeter (s)
The semi-perimeter s is half of the perimeter:
\[ s = \frac{84}{2} = 42 \text{ cm} \]

### Step 5: 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 we have:
- \( s = 42 \)
- \( a = 26 \)
- \( b = 28 \)
- \( c = 30 \)

Calculating the terms:
- \( s - a = 42 - 26 = 16 \)
- \( s - b = 42 - 28 = 14 \)
- \( s - c = 42 - 30 = 12 \)

Now substituting these into Heron's formula:
\[ A = \sqrt{42 \times 16 \times 14 \times 12} \]

### Step 6: Simplify the expression
Calculating inside the square root:
- First, calculate \( 42 \times 16 = 672 \)
- Next, calculate \( 14 \times 12 = 168 \)
- Now, multiply these results: \( 672 \times 168 \)

Calculating \( 672 \times 168 \):
\[ 672 \times 168 = 112896 \]

Now, take the square root:
\[ A = \sqrt{112896} = 336 \text{ cm}^2 \]

### Final Answer
The area of the triangle is \( 336 \text{ cm}^2 \).

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