Two sides of a triangular field are 85 m and 154 m in length and its perimeter is 324 m. Find (i) the area of the field and (ii) the length of the perpendicular from the opposite vertex on the side measuring 154 m.

Answer

To solve the problem step by step, we will find the area of the triangular field using Heron's formula and then calculate the length of the perpendicular from the opposite vertex to the side measuring 154 m.

### Step 1: Identify the sides of the triangle
Given:
- Side A = 85 m
- Side B = 154 m
- Perimeter = 324 m

To find the third side (C), we use the perimeter:
\[ C = \text{Perimeter} - (A + B) \]
\[ C = 324 - (85 + 154) \]
\[ C = 324 - 239 = 85 \text{ m} \]

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter (s) is half of the perimeter:
\[ s = \frac{A + B + C}{2} \]
\[ s = \frac{85 + 154 + 85}{2} \]
\[ s = \frac{324}{2} = 162 \text{ m} \]

### Step 3: Apply Heron's formula to find the area (A)
Heron's formula for the area of a triangle is given by:
\[ \text{Area} = \sqrt{s \cdot (s - A) \cdot (s - B) \cdot (s - C)} \]

Substituting the values:
\[ \text{Area} = \sqrt{162 \cdot (162 - 85) \cdot (162 - 154) \cdot (162 - 85)} \]
\[ = \sqrt{162 \cdot 77 \cdot 8 \cdot 77} \]
\[ = \sqrt{162 \cdot 77^2 \cdot 8} \]

### Step 4: Simplify the area calculation
Calculating \( 77^2 \):
\[ 77^2 = 5929 \]

Now substituting back:
\[ \text{Area} = \sqrt{162 \cdot 5929 \cdot 8} \]
\[ = \sqrt{162 \cdot 47432} \]
\[ = \sqrt{7693440} \]

Calculating \( \sqrt{7693440} \):
\[ \text{Area} \approx 2772 \text{ m}^2 \]

### Step 5: Find the length of the perpendicular (h) from the opposite vertex to the base (154 m)
Using the area formula for a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Where the base is 154 m and the area is 2772 m²:
\[ 2772 = \frac{1}{2} \times 154 \times h \]

### Step 6: Solve for height (h)
Rearranging the equation:
\[ h = \frac{2 \times 2772}{154} \]
\[ h = \frac{5544}{154} \]
\[ h \approx 36 \text{ m} \]

### Final Answers
(i) The area of the triangular field is approximately **2772 m²**.
(ii) The length of the perpendicular from the opposite vertex on the side measuring 154 m is **36 m**.