If the sides of a triangle are 50 m, 78 m and 112 m, then the perpendicular distance from the opposite angle on the side 112 m is

Options

A. 10 m
B. 30 m
C. 5 m
D. 20 m

Answer

To find the perpendicular distance from the opposite angle to the side of length 112 m in a triangle with sides 50 m, 78 m, and 112 m, we can use Heron's formula to first calculate the area of the triangle and then use the area to find the height (perpendicular distance).

### Step-by-Step Solution:

1. **Identify the sides of the triangle:**
- Let \( a = 50 \, \text{m} \)
- Let \( b = 78 \, \text{m} \)
- Let \( c = 112 \, \text{m} \)

2. **Calculate the semi-perimeter (s):**
\[ s = \frac{a + b + c}{2} = \frac{50 + 78 + 112}{2} = \frac{240}{2} = 120 \, \text{m} \]

3. **Apply Heron's formula to find the area (A):**
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substitute the values:
\[ A = \sqrt{120(120-50)(120-78)(120-112)} \]
\[ = \sqrt{120 \times 70 \times 42 \times 8} \]

4. **Calculate each term:**
- \( s - a = 120 - 50 = 70 \)
- \( s - b = 120 - 78 = 42 \)
- \( s - c = 120 - 112 = 8 \)

5. **Now calculate the area:**
\[ A = \sqrt{120 \times 70 \times 42 \times 8} \]
- First, calculate \( 120 \times 70 = 8400 \)
- Then, \( 8400 \times 42 = 352800 \)
- Finally, \( 352800 \times 8 = 2822400 \)
- Thus, \( A = \sqrt{2822400} \)

6. **Find the square root:**
- \( A = 1680 \, \text{m}^2 \)

7. **Use the area to find the height (h):**
The area of the triangle can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( c = 112 \, \text{m} \):
\[ 1680 = \frac{1}{2} \times 112 \times h \]

8. **Solve for height (h):**
\[ 1680 = 56h \]
\[ h = \frac{1680}{56} = 30 \, \text{m} \]

### Final Answer:
The perpendicular distance from the opposite angle to the side of length 112 m is **30 m**.