What is the ratio of side and height of an equilateral triangle?

Options

A. 2:1
B. 1:1
C. `2:sqrt3`
D. `sqrt3:2`

Answer

To find the ratio of the side to the height of an equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Side Length**:
Let the side length of the equilateral triangle be denoted as \( s \). For simplicity, we can assume \( s = 2A \) (where \( A \) is a constant).

2. **Identify the Height**:
In an equilateral triangle, the height can be calculated using the Pythagorean theorem. When we drop a perpendicular from one vertex to the opposite side, it bisects the side into two equal parts. Therefore, each half of the side will be \( A \).

3. **Apply the Pythagorean Theorem**:
In triangle \( ABD \) (where \( D \) is the midpoint of side \( BC \)):
- \( AB = 2A \) (the side of the triangle)
- \( AD \) is the height we want to find
- \( BD = A \) (half of the side)

According to the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ (2A)^2 = AD^2 + A^2 \]
This simplifies to:
\[ 4A^2 = AD^2 + A^2 \]

4. **Solve for Height \( AD \)**:
Rearranging the equation gives:
\[ AD^2 = 4A^2 - A^2 \]
\[ AD^2 = 3A^2 \]
Taking the square root of both sides:
\[ AD = \sqrt{3A^2} = A\sqrt{3} \]

5. **Calculate the Ratio**:
Now we have the side \( s = 2A \) and the height \( AD = A\sqrt{3} \). The ratio of the side to the height is:
\[ \text{Ratio} = \frac{\text{Side}}{\text{Height}} = \frac{2A}{A\sqrt{3}} = \frac{2}{\sqrt{3}} \]

6. **Simplify the Ratio**:
To express the ratio in a more standard form, we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

### Final Answer:
The ratio of the side to the height of an equilateral triangle is \( 2 : \sqrt{3} \).