What is the ratio of side to the 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 1: Define the side length of the equilateral triangle
Let the side length of the equilateral triangle be denoted as \( a \) cm.

### Step 2: Determine the height of the equilateral triangle
The height \( h \) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} \times a \]

### Step 3: Set up the ratio of the side to the height
We need to find the ratio of the side to the height, which can be expressed as:
\[ \text{Ratio} = \frac{\text{Side}}{\text{Height}} = \frac{a}{h} \]

### Step 4: Substitute the height in the ratio
Substituting the expression for height into the ratio gives:
\[ \text{Ratio} = \frac{a}{\frac{\sqrt{3}}{2} \times a} \]

### Step 5: Simplify the ratio
Since \( a \) is present in both the numerator and the denominator, we can cancel it out:
\[ \text{Ratio} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 6: Express the ratio in standard form
The ratio can be expressed as:
\[ \text{Ratio} = 2 : \sqrt{3} \]

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

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