`ABC` is an equilateral triangle of side `2a`. Find each of its altitudes.

Options

A. `asqrt3`
B. `asqrt2`
C. `2asqrt3`
D. None

Explanation

To find the altitude of an equilateral triangle \( ABC \) with side length \( 2a \), we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
In an equilateral triangle, all sides are equal, and all angles are \( 60^\circ \). The altitude divides the triangle into two 30-60-90 right triangles.

### Step 2: Set up the triangle and the altitude
Let \( AD \) be the altitude from vertex \( A \) to side \( BC \). Since \( ABC \) is an equilateral triangle, we know:
- \( AB = AC = BC = 2a \)

### Step 3: Determine the lengths of segments
The altitude \( AD \) divides \( BC \) into two equal segments:
- \( BD = DC = a \)

### Step 4: Use the Pythagorean theorem
In right triangle \( ABD \):
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ (2a)^2 = AD^2 + a^2 \]

### Step 5: Simplify the equation
Calculating \( (2a)^2 \):
\[ 4a^2 = AD^2 + a^2 \]
Now, isolate \( AD^2 \):
\[ AD^2 = 4a^2 - a^2 = 3a^2 \]

### Step 6: Solve for \( AD \)
Taking the square root of both sides:
\[ AD = \sqrt{3a^2} = a\sqrt{3} \]

### Step 7: Conclude the result
Since \( ABC \) is equilateral, the altitudes \( AD \), \( BE \), and \( CF \) are all equal:
\[ AD = BE = CF = a\sqrt{3} \]

### Final Answer
Each altitude of the equilateral triangle \( ABC \) is \( a\sqrt{3} \).

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