What is the area of a triangle having perimeter 32cm, one side 11cm and difference of other two sides 5cm?

Options

A. `8sqrt(30)cm^(2)`
B. `5sqrt(35) cm^(2)`
C. `6sqrt(30) cm^(2)`
D. `8sqrt(2) cm^(2)`

Answer

To find the area of the triangle with the given conditions, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be \( a \), \( b \), and \( c \). We know:
- One side \( c = 11 \, \text{cm} \)
- The perimeter of the triangle is \( 32 \, \text{cm} \)
- The difference between the other two sides \( a \) and \( b \) is \( 5 \, \text{cm} \)

### Step 2: Set up equations
From the perimeter, we have:
\[ a + b + c = 32 \]
Substituting \( c = 11 \):
\[ a + b + 11 = 32 \]
Thus,
\[ a + b = 32 - 11 = 21 \quad \text{(Equation 1)} \]

From the difference of the sides, we have:
\[ a - b = 5 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we can solve these two equations simultaneously. We can add Equation 1 and Equation 2:
\[ (a + b) + (a - b) = 21 + 5 \]
This simplifies to:
\[ 2a = 26 \]
Thus,
\[ a = \frac{26}{2} = 13 \, \text{cm} \]

Now, substitute \( a \) back into Equation 1 to find \( b \):
\[ 13 + b = 21 \]
So,
\[ b = 21 - 13 = 8 \, \text{cm} \]

Now we have the sides of the triangle:
- \( a = 13 \, \text{cm} \)
- \( b = 8 \, \text{cm} \)
- \( c = 11 \, \text{cm} \)

### Step 4: Calculate the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} = \frac{32}{2} = 16 \, \text{cm} \]

### Step 5: Use Heron's formula to find the area
Heron's formula states that the area \( A \) of a triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ A = \sqrt{16(16-13)(16-8)(16-11)} \]
Calculating each term:
\[ A = \sqrt{16 \times 3 \times 8 \times 5} \]

### Step 6: Simplify the expression
Calculating the products inside the square root:
\[ 16 \times 3 = 48 \]
\[ 48 \times 8 = 384 \]
\[ 384 \times 5 = 1920 \]
So,
\[ A = \sqrt{1920} \]

### Step 7: Further simplify \( \sqrt{1920} \)
Breaking down \( 1920 \):
\[ 1920 = 64 \times 30 = 8^2 \times 30 \]
Thus,
\[ A = 8 \sqrt{30} \]

### Step 8: Final area calculation
Calculating \( 8 \sqrt{30} \):
Using an approximate value for \( \sqrt{30} \approx 5.477 \):
\[ A \approx 8 \times 5.477 \approx 43.816 \, \text{cm}^2 \]

### Conclusion
The area of the triangle is approximately \( 43.82 \, \text{cm}^2 \).

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