A triangular field BAD, right angled at A has `AB=180m` and `/_DBA=30^@`. The length AD is

Options

A. ` 29 sqrt (3)`
B. ` 38 sqrt(3) m `
C. `43 sqrt(3)` m
D. ` 60 sqrt( 3) m `

Answer

To find the length of AD in the triangular field BAD, we can use trigonometric ratios. Here's the step-by-step solution:

### Step 1: Understand the Triangle
We have a right-angled triangle BAD where:
- Angle A is the right angle.
- AB = 180 m (the base).
- Angle DBA = 30°.

### Step 2: Identify the Sides
In triangle BAD:
- AB is the base (adjacent side to angle DBA).
- AD is the height (opposite side to angle DBA).

### Step 3: Use the Tangent Function
We can use the tangent function, which is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For angle DBA (30°):
\[ \tan(30°) = \frac{AD}{AB} \]

### Step 4: Substitute Known Values
We know:
- \(\tan(30°) = \frac{1}{\sqrt{3}}\)
- \(AB = 180 m\)

Substituting these values into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{AD}{180} \]

### Step 5: Solve for AD
To find AD, we can cross-multiply:
\[ AD = 180 \times \frac{1}{\sqrt{3}} \]
\[ AD = \frac{180}{\sqrt{3}} \]

### Step 6: Rationalize the Denominator
To simplify \(\frac{180}{\sqrt{3}}\), we multiply the numerator and the denominator by \(\sqrt{3}\):
\[ AD = \frac{180 \sqrt{3}}{3} = 60 \sqrt{3} \text{ m} \]

### Final Answer
Thus, the length of AD is:
\[ AD = 60 \sqrt{3} \text{ m} \]

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