In a triangular field having sides 30 m, 72 m and 78 m, the length of the altitude to the side measuring 72 m is

Options

A. 25 m
B. 28 m
C. 30 m
D. 35 m

Answer

To find the length of the altitude to the side measuring 72 m in a triangular field with sides 30 m, 72 m, and 78 m, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 30 \, \text{m} \)
- \( b = 72 \, \text{m} \)
- \( c = 78 \, \text{m} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{30 + 72 + 78}{2} = \frac{180}{2} = 90 \, \text{m} \]

### Step 3: Use Heron's formula to find the area (A) of the triangle
Heron's formula states:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{90(90 - 30)(90 - 72)(90 - 78)} \]
Calculating the differences:
\[ A = \sqrt{90 \times 60 \times 18 \times 12} \]

### Step 4: Simplify the area calculation
Calculating step by step:
- First calculate \( 60 \times 18 = 1080 \)
- Then calculate \( 1080 \times 12 = 12960 \)
- Finally, calculate \( 90 \times 12960 = 1166400 \)

Now, take the square root:
\[ A = \sqrt{1166400} = 1080 \, \text{m}^2 \]

### Step 5: Relate the area to the altitude
The area of the triangle can also be expressed in terms of the base and height (altitude):
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is 72 m and the height is \( h \):
\[ 1080 = \frac{1}{2} \times 72 \times h \]

### Step 6: Solve for the height (h)
Rearranging the equation to solve for \( h \):
\[ 1080 = 36h \quad \Rightarrow \quad h = \frac{1080}{36} = 30 \, \text{m} \]

### Final Answer
The length of the altitude to the side measuring 72 m is:
\[ \boxed{30 \, \text{m}} \]