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 follow these steps:

### Step 1: Calculate the semi-perimeter (s) of the triangle.
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 50 \, m \), \( b = 78 \, m \), and \( c = 112 \, m \).

\[ s = \frac{50 + 78 + 112}{2} = \frac{240}{2} = 120 \, m \]

### Step 2: Calculate the area (A) of the triangle using Heron's formula.
Heron's formula for the area of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ A = \sqrt{120 \times (120 - 50) \times (120 - 78) \times (120 - 112)} \]
Calculating each term:
\[ A = \sqrt{120 \times 70 \times 42 \times 8} \]

### Step 3: Simplify the expression under the square root.
Calculating the product:
\[ 120 \times 70 = 8400 \]
\[ 8400 \times 42 = 352800 \]
\[ 352800 \times 8 = 2822400 \]
Thus,
\[ A = \sqrt{2822400} \]

### Step 4: Calculate the square root.
To simplify \( \sqrt{2822400} \):
\[ 2822400 = 1680^2 \]
Thus,
\[ A = 1680 \, m^2 \]

### Step 5: Use the area to find the perpendicular height (h) from the opposite angle to the base of length 112 m.
The area of a triangle can also be expressed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is 112 m, and we need to find the height \( h \):
\[ 1680 = \frac{1}{2} \times 112 \times h \]

### Step 6: Solve for \( h \).
Rearranging the equation:
\[ h = \frac{1680 \times 2}{112} = \frac{3360}{112} = 30 \, m \]

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

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