In a triangle ABC, If a perpendicular AD is drawn from point A to side BC then find the value of `angle A` if `AB = 6 sqrt3 cm, CD = 3 sqrt3 cm and AD = 9cm`

Options

A. 45°
B. 90°
C. 30°
D. 60°

Answer

To find the value of angle A in triangle ABC, where a perpendicular AD is drawn from point A to side BC, we will follow these steps:

### Step 1: Understand the Triangle Configuration
We have triangle ABC with:
- AB = 6√3 cm
- CD = 3√3 cm (where D is the foot of the perpendicular from A to BC)
- AD = 9 cm (the height from A to BC)

### Step 2: Identify Right Triangles
Since AD is perpendicular to BC, we can analyze two right triangles: ABD and ADC.

### Step 3: Analyze Triangle ABD
In triangle ABD:
- AD is the perpendicular (height) = 9 cm
- AB is the hypotenuse = 6√3 cm

Let angle ADB be θ. We can use the sine function to find θ:
\[ \sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AD}{AB} = \frac{9}{6\sqrt{3}} \]

### Step 4: Simplify the Sine Expression
\[ \sin(θ) = \frac{9}{6\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \]

### Step 5: Find Angle θ
The sine value of \(\frac{\sqrt{3}}{2}\) corresponds to:
\[ θ = 60^\circ \]

### Step 6: Analyze Triangle ADC
In triangle ADC:
- AD is the perpendicular = 9 cm
- CD is the base = 3√3 cm

Let angle ADC be α. We can use the tangent function to find α:
\[ \tan(α) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AD}{CD} = \frac{9}{3\sqrt{3}} \]

### Step 7: Simplify the Tangent Expression
\[ \tan(α) = \frac{9}{3\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 8: Find Angle α
The tangent value of \(\sqrt{3}\) corresponds to:
\[ α = 60^\circ \]

### Step 9: Calculate Angle A
Now, we know:
- Angle ADB (θ) = 60°
- Angle ADC (α) = 60°

Let angle A (the angle at point A) be β. The sum of angles in triangle ABC is:
\[ β + θ + α = 180^\circ \]
Substituting the known angles:
\[ β + 60^\circ + 60^\circ = 180^\circ \]
\[ β + 120^\circ = 180^\circ \]
\[ β = 180^\circ - 120^\circ = 60^\circ \]

### Final Answer
Thus, the value of angle A is:
\[ \boxed{60^\circ} \]