For a triangle, base is `6sqrt(3)` cm and two base angles are `30^@` and `60^@`. Then height of the triangle is

Options

A. `3sqrt(3) cm `
B. `4.5 cm`
C. `4sqrt(3) cm`
D. `2sqrt(3) cm`

Answer

To find the height of the triangle with a base of \(6\sqrt{3}\) cm and base angles of \(30^\circ\) and \(60^\circ\), we can follow these steps:

### Step 1: Identify the triangle type
Since the triangle has angles of \(30^\circ\), \(60^\circ\), and \(90^\circ\), it is a right triangle.

**Hint:** Remember that the angles in a triangle add up to \(180^\circ\).

### Step 2: Set up the triangle
Let the triangle be labeled as \(ABC\) where:
- \(A\) is the vertex opposite the hypotenuse,
- \(B\) is the vertex where the \(30^\circ\) angle is located,
- \(C\) is the vertex where the \(60^\circ\) angle is located.

### Step 3: Use the properties of a \(30^\circ-60^\circ-90^\circ\) triangle
In a \(30^\circ-60^\circ-90^\circ\) triangle, the sides are in the ratio:
- Opposite \(30^\circ\) : Opposite \(60^\circ\) : Hypotenuse = \(1 : \sqrt{3} : 2\).

### Step 4: Identify the hypotenuse
The hypotenuse \(AC\) is opposite the right angle, and in this case, it is the side opposite the \(90^\circ\) angle. Since the base \(BC\) is given as \(6\sqrt{3}\) cm, we can use this to find the other sides.

### Step 5: Calculate the sides
Let:
- \(BC\) (base) = \(6\sqrt{3}\) cm (opposite \(60^\circ\)),
- \(AB\) (height) = height we need to find (opposite \(30^\circ\)),
- \(AC\) (hypotenuse) = \(2 \times AB\).

Using the ratio:
- The side opposite \(30^\circ\) (height \(AB\)) is half of the hypotenuse, and the side opposite \(60^\circ\) (base \(BC\)) is \(\sqrt{3}\) times the height.

### Step 6: Set up the equations
From the properties:
1. \(BC = AB \cdot \sqrt{3}\)
2. \(AC = 2 \cdot AB\)

### Step 7: Substitute and solve
From the first equation:
\[ 6\sqrt{3} = AB \cdot \sqrt{3} \]
Dividing both sides by \(\sqrt{3}\):
\[ AB = 6 \text{ cm} \]

### Step 8: Calculate the height
Thus, the height \(AB\) is:
\[ AB = 6 \text{ cm} \]

### Final Answer
The height of the triangle is \(6\) cm.

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