In a scalene triangle ,one side exceeds the other two sides by 4 cm and 5 cm , respectively , and the perimeter of the triangle is 36 cm .Find the area of the triangle in `cm^(2)` .

Options

A. `63`
B. `9sqrt(10)`
C. `18sqrt(10)`
D. `12sqrt(21)`

Answer

To solve the problem step by step, we will first define the sides of the triangle based on the given conditions, then calculate the sides, and finally find the area using Heron's formula.

### Step 1: Define the sides of the triangle
Let the three sides of the scalene triangle be:
- Side A = x (the smallest side)
- Side B = y (the second side)
- Side C = x + 4 (the side that exceeds side A by 4 cm)
- Side D = y + 5 (the side that exceeds side B by 5 cm)

### Step 2: Set up the perimeter equation
According to the problem, the perimeter of the triangle is 36 cm. Therefore, we can write the equation:
\[ x + y + (x + 4) + (y + 5) = 36 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 2x + 2y + 9 = 36 \]
Now, subtract 9 from both sides:
\[ 2x + 2y = 27 \]
Divide the entire equation by 2:
\[ x + y = 13.5 \] (Equation 1)

### Step 4: Set up the second equation
From the definitions of the sides, we can also express:
\[ C = x + 4 \]
\[ D = y + 5 \]

Since we know that \( C \) and \( D \) are also sides of the triangle, we can set up another equation:
\[ (x + 4) + (y + 5) + x = 36 \]
This simplifies to:
\[ 2x + y + 9 = 36 \]
Subtracting 9 from both sides gives:
\[ 2x + y = 27 \] (Equation 2)

### Step 5: Solve the system of equations
Now we have two equations:
1. \( x + y = 13.5 \)
2. \( 2x + y = 27 \)

We can solve for \( y \) in terms of \( x \) from Equation 1:
\[ y = 13.5 - x \]

Substituting \( y \) in Equation 2:
\[ 2x + (13.5 - x) = 27 \]
This simplifies to:
\[ 2x + 13.5 - x = 27 \]
\[ x + 13.5 = 27 \]
Subtracting 13.5 from both sides:
\[ x = 13.5 - 27 \]
\[ x = 13.5 - 13.5 \]
\[ x = 13.5 \]

Now substituting \( x \) back to find \( y \):
\[ y = 13.5 - 13.5 = 0 \]

### Step 6: Find the lengths of the sides
Now we can find the lengths of the sides:
- Side A (x) = 11 cm
- Side B (y) = 10 cm
- Side C = x + 4 = 15 cm

### Step 7: Calculate the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{A + B + C}{2} = \frac{11 + 10 + 15}{2} = \frac{36}{2} = 18 \]

### Step 8: Use Heron's formula to find the area
Using Heron's formula:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ \text{Area} = \sqrt{18(18 - 11)(18 - 10)(18 - 15)} \]
\[ = \sqrt{18 \times 7 \times 8 \times 3} \]
\[ = \sqrt{18 \times 168} \]

### Step 9: Simplify the area calculation
Calculating:
\[ = \sqrt{3024} \]
Breaking it down:
\[ = 12 \sqrt{21} \]

### Final Answer
Thus, the area of the triangle is:
\[ \text{Area} = 12 \sqrt{21} \, \text{cm}^2 \]