What will be the length of the altitude of an equilateral triangle of side 2a cm?

Options

A. a
B. `sqrt3a`
C. `a/2`
D. `sqrt3/2a`

Answer

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

### Step 1: Understand the properties of an equilateral triangle
In an equilateral triangle, all sides are equal, and the altitude (height) also acts as the median and angle bisector.

### Step 2: Draw the triangle and label it
Let’s denote the equilateral triangle as \( \triangle ABC \) where \( AB = BC = CA = 2a \) cm. We will drop a perpendicular from point \( A \) to the midpoint \( D \) of side \( BC \).

### Step 3: Determine the length of \( BD \)
Since \( D \) is the midpoint of \( BC \), we can find the length of \( BD \):
\[ BD = \frac{BC}{2} = \frac{2a}{2} = a \text{ cm} \]

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

### Step 5: Simplify the equation
Now, we simplify this equation:
\[ 4a^2 = AD^2 + a^2 \]
Subtract \( a^2 \) from both sides:
\[ 4a^2 - a^2 = AD^2 \]
\[ 3a^2 = AD^2 \]

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

### Conclusion
Thus, the length of the altitude \( AD \) of the equilateral triangle is:
\[ AD = a\sqrt{3} \text{ cm} \]