Find the area of a triangular field whose sides are 91 m, 98 m and 105 m in length. Find the height corresponding to the longest side.

Answer

To find the area of a triangular field with sides measuring 91 m, 98 m, and 105 m, and to determine the height corresponding to the longest side, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 91 \, \text{m} \)
- \( b = 98 \, \text{m} \)
- \( c = 105 \, \text{m} \) (the longest side)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{91 + 98 + 105}{2} = \frac{294}{2} = 147 \, \text{m} \]

### Step 3: Use Heron's formula to find the area (A)
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:
- \( s - a = 147 - 91 = 56 \)
- \( s - b = 147 - 98 = 49 \)
- \( s - c = 147 - 105 = 42 \)

Now substituting into the formula:
\[ A = \sqrt{147 \times 56 \times 49 \times 42} \]

### Step 4: Calculate the area
Calculating the product:
1. Calculate \( 147 \times 56 = 8232 \)
2. Calculate \( 49 \times 42 = 2058 \)
3. Now multiply these two results:
\[ 8232 \times 2058 = 16912296 \]
4. Finally, take the square root:
\[ A = \sqrt{16912296} \approx 4116 \, \text{m}^2 \]

### Step 5: Find the height corresponding to the longest side
To find the height \( h \) corresponding to the longest side \( c = 105 \, \text{m} \), we use the area formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( c \):
\[ 4116 = \frac{1}{2} \times 105 \times h \]
Rearranging to find \( h \):
\[ h = \frac{4116 \times 2}{105} = \frac{8232}{105} \approx 78.4 \, \text{m} \]

### Final Answers
- Area of the triangular field: \( 4116 \, \text{m}^2 \)
- Height corresponding to the longest side: \( 78.4 \, \text{m} \)