The base of triangle is `12sqrt(3)` cm and two angles at the base are `30^(@)and60^(@)` respectively .The altitude of the triangle is

Options

A. 12 cm
B. 6 cm
C. `10sqrt(3)`cm
D. 9 cm

Answer

To find the altitude of the triangle with a base of \(12\sqrt{3}\) cm and angles of \(30^\circ\) and \(60^\circ\) at the base, we can follow these steps:

### Step 1: Understand the Triangle Configuration
We have a triangle ABC where:
- Angle A = \(30^\circ\)
- Angle B = \(60^\circ\)
- Angle C = \(90^\circ\) (since the angles at the base add up to \(90^\circ\))

### Step 2: Identify the Base and Height
The base of the triangle (BC) is given as \(12\sqrt{3}\) cm. We need to find the altitude (height) from point A to line BC.

### Step 3: Use Trigonometric Ratios
In triangle ABC, we can use the properties of a right triangle. The sides opposite the angles \(30^\circ\) and \(60^\circ\) have specific ratios:
- The side opposite \(30^\circ\) is half the hypotenuse.
- The side opposite \(60^\circ\) is \(\sqrt{3}\) times the side opposite \(30^\circ\).

### Step 4: Determine the Lengths of the Sides
Let:
- The length of side AC (opposite \(60^\circ\)) = \(x\)
- The length of side AB (opposite \(30^\circ\)) = \(x/2\)

From the triangle properties:
- The base BC can be expressed as:
\[ BC = AC + AB = x + \frac{x}{2} = \frac{3x}{2} \]
Since \(BC = 12\sqrt{3}\):
\[ \frac{3x}{2} = 12\sqrt{3} \]

### Step 5: Solve for x
To find \(x\), multiply both sides by \(2\):
\[ 3x = 24\sqrt{3} \]
Now divide by \(3\):
\[ x = 8\sqrt{3} \]

### Step 6: Find the Height
Now, we can find the height (altitude) from point A to line BC. The height (h) can be calculated using the side opposite \(30^\circ\):
\[ h = \frac{x}{2} = \frac{8\sqrt{3}}{2} = 4\sqrt{3} \]

### Final Answer
The altitude of the triangle is \(4\sqrt{3}\) cm.

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