Two equilateral triangles of side `8sqrt3` cm are joined to form a quadrilateral. What is the altitude of the quadrilateral?

Options

A. 12 cm
B. 14 cm
C. 16 cm
D. 15 cm

Answer

To find the altitude of the quadrilateral formed by joining two equilateral triangles with a side length of \(8\sqrt{3}\) cm, we can follow these steps:

### Step 1: Understand the shape formed
When two equilateral triangles are joined, they form a rhombus. This is because the two triangles share a common side, and the opposite sides are parallel and equal in length.

### Step 2: Calculate the altitude of one equilateral triangle
The formula for the altitude (height) \(h\) of an equilateral triangle with side length \(a\) is given by:

\[ h = \frac{\sqrt{3}}{2} \times a \]

### Step 3: Substitute the side length into the formula
Here, the side length \(a\) is \(8\sqrt{3}\) cm. Substituting this value into the formula gives:

\[ h = \frac{\sqrt{3}}{2} \times (8\sqrt{3}) \]

### Step 4: Simplify the expression
Now, we can simplify the expression:

\[ h = \frac{\sqrt{3}}{2} \times 8\sqrt{3} = \frac{8 \times 3}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 5: Conclusion
Since the altitude of the quadrilateral formed by the two equilateral triangles is the same as the altitude of one of the triangles, the altitude of the quadrilateral is:

\[ \text{Altitude of the quadrilateral} = 12 \text{ cm} \]

### Final Answer
The altitude of the quadrilateral is **12 cm**.

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